To make the pasta:

1. To bring the dough together using a machine: In a stand mixer fitted with the paddle attachment, on low speed, combine the flour, eggs, olive oil, and salt and mix until crumbly.
2. If using a food processor, pulse together the flour, eggs, olive oil, and salt until the dough starts coming together.
3. Incrementally, add 1 teaspoon water at a time until the dough becomes a craggy ball and there are barely any dry crumbs of flour remaining. (Some days, it’s 3 teaspoons; other days, it’s a little more. Just don’t add too much water, since the flour will continue to hydrate as the dough is kneaded by hand.)
4. Lightly flour a work surface, place the dough on the surface, and knead until it becomes a smooth and supple oval ball and springs back when lightly pressed, about 10 minutes. Remove the crease line in the dough by pinching the crease together and pushing down on it with your palm.
5. Cover the ball of dough with plastic wrap, or turn a mixing bowl upside down and place it over the dough on the work surface, and let rest on the counter for 30 to 45 minutes. After resting, the dough may be sticky; if so, dust it with a little flour.
6. If rolling pasta by machine: Divide the dough into four equal pieces. (If using a hand-cranked pasta roller, it is easier to work by dividing into six pieces.) Work with one piece at a time, keeping the remaining dough well covered in plastic wrap. Flatten the piece of dough between your palms before feeding it through the pasta machine once on the widest setting (#1).
7. Still on the widest setting and at low speed, feed the dough through the pasta roller, then laminate it by folding it in thirds, just as you would fold a letter to fit into an envelope. Repeat this step a handful of times, turning the dough 90 degrees between each pass. Keep your folds tidy in order to create an even rectangular sheet.
8. Now, feed the sheet of dough through the rollers, turning the knob down incrementally to a narrower setting between each pass (there’s no need to fold between passes anymore), until it’s rolled to the desired thickness. If the dough feels sticky or puckers as it’s rolled, sprinkle it lightly with flour and use your hand to spread it gently across the sheet before feeding it through the machine again. This helps to keep the dough flat and smooth and prevents it from getting stuck.
9. Once the pasta is cut or filled, place on a flour-dusted baking sheet. Lightly dust a little more flour on top of the pasta so it doesn’t stick. (It’s completely okay if the pasta dries out at this stage.)
10. If cooking within a couple of hours, simply leave on the counter. You can place stuffed pasta, uncovered, in the refrigerator for a few hours but be very careful that they don’t become a sticky mess.
11. To store stuffed or cut pasta, freeze in a single layer on a baking sheet, transfer the frozen pasta to a ziplock bag or airtight container, and freeze for about 1 month. For longer pasta, like fettucine, make small individual portioned nests before freezing.

To prep the veggies:

1. Preheat the oven to 350°F. Line two large baking sheets with parchment paper.
2. Toss the squash in 1/4 cup of the olive oil, season with salt and pepper, and spread out evenly in a single layer on the prepared baking sheets.
3. Scatter 2 tablespoons of the butter and 2 tablespoons of the water on each sheet, cover tightly with aluminum foil, and cook until softened, about 15 minutes, then remove from the oven. Set aside.
4. While the squash is cooking, in a large frying pan over medium-high heat, warm 2 tablespoons butter and 2 tablespoons olive oil with 3 garlic cloves.
5. Working in batches if needed, add the cauliflower, season with a little salt, and sauté, basting with the garlic-infused butter. Try not to break up the delicate slices (adding more butter and olive oil as needed). Cook until tender, about 15 minutes. Remove everything from the pan and set aside.
6. Add the remaining 2 tablespoons butter, remaining 2 tablespoons olive oil, and remaining 3 garlic cloves to the same pan. Working in batches if needed, add the fennel and season with a little salt. Sauté until soft and caramelized, 15 to 20 minutes (adding more butter and oil as needed).
7. Turn off the heat and remove all the garlic from both the fennel and cauliflower. Finely chop the garlic and set aside.

To make the béchamel:

1. In a large heavy-bottomed saucepan over low heat, melt the butter. Add the flour and cook for 2 minutes while constantly whisking. Gradually whisk in the milk, until smooth.
2. When the béchamel has the consistency of heavy cream, whisk in the chopped garlic, Parmigiano, salt, nutmeg, and some cranks of pepper to combine.
3. Bring a large pot of lightly salted water to a boil. Working in batches, add four sheets of pasta at a time and blanch just until softened, 1 to 2 minutes, or cook for 7 to 8 minutes or according to package instructions if using store-bought.
4. Using a large spider, remove the sheets and lay them out on a clean surface. (Careful, they’ll be hot!)
5. When cool enough to touch, unfurl the sheets and splash a little olive oil over them to keep them from sticking. Repeat with the remaining pasta sheets.

To assemble:

1. Preheat the oven to 400°F.
2. Butter a deep 9 by 13-inch baking dish, spread a thin layer of béchamel over the bottom, cover with a layer of pasta, then add a layer of squash.
3. Sprinkle some of the fennel and Parmigiano on top, dot with a little pesto, and season with pepper.
4. Cover with another layer of pasta, béchamel, and now cauliflower. Sprinkle more fennel and Parmigiano on top and add a little more pesto and pepper.
5. Finish with a layer of squash, since you’ll still have some left, then a final layer of pasta, béchamel, Parmigiano, and basil. Scatter the butter pieces on top.
6. Bake the lasagna until the top is golden and the edges are crisp, about 30 minutes. Let it rest and compose itself for about 10 minutes and sprinkle with sea salt just before serving.