Carbonara Lasagna

Place a rack in lower third of oven; preheat to 375°. Lightly butter an 11x8" (2-qt.) glass or ceramic baking dish. Melt 1 Tbsp. butter in a large skillet over medium-high heat. Add pancetta and cook, stirring occasionally, until most of the fat is rendered and pancetta is golden, 8–10 minutes. Reduce heat to medium, add shallots, and cook, stirring frequently, until softened, about 5 minutes. Using a slotted spoon, transfer shallots and pancetta to a large bowl. Let cool slightly, then stir in ricotta, Fontina, and Parmesan.

Pour off fat and return skillet to medium-high heat. Add thyme, then pour in cream, scraping up browned bits stuck to bottom of pan. Bring to a simmer; season with salt and pepper. Remove from heat.

If you're using fresh pasta, cut noodles to fit pan and arrange a layer of 2 sheets in the bottom, overlapping slightly. (The fresh pasta is bigger and wider than the grocery store dry kind, so 2 noodles constitutes a whole layer. If you’re using dry noodles, use enough noodles in each layer so they overlap slightly and cover bottom of pan.) Dollop with one-quarter of cheese mixture and top with another layer of noodles. Repeat three more times with the remaining filling and noodles, ending with a layer of noodles (you'll have several leftover). Pour cream mixture over. Shake pan gently to distribute and press with a spatula to compact slightly. Bake lasagna until golden, about 40 minutes.

Crack eggs into individual bowls. Press a 2" round cookie cutter into lasagna, going about halfway down. Carefully remove top layers of lasagna to create a well. Repeat to create 3 more. Gently drop 1 egg in each well. Cover with foil and continue to bake until egg whites are just set, 5–7 minutes longer. Let sit 15 minutes before serving.

Pulse eggs, oil, 2 cups flour, and 1 tsp. salt in a food processor in 1-second intervals until dough is moistened and comes together in small beads that resemble couscous. Don’t let dough form a ball.

Turn dough out onto a work surface and knead (without adding flour) until smooth and elastic, 2–3 minutes (it will be almost too stiff to knead). Wrap dough in plastic and let sit at room temperature at least 30 minutes or up to 1 hour (the dough will soften and be easy to work with after sitting).

Arrange several clean kitchen towels on a work surface and dust lightly with flour. Cut dough into quarters and cover 3 pieces with a damp towel. Flatten 1 piece of dough and dust lightly with flour. Roll dough through a pasta machine, starting on the lowest (widest) setting. Fold rolled dough into thirds like a letter, then roll it through again, feeding the open, less wide end (the side where you can see the fold) through the machine first. Repeat 3 times, then start to roll dough using thinner settings, folding it and putting it through the same setting 2 times before progressing to the next thinner setting, and flouring dough as needed to keep from sticking. Work your way through to the second to last setting (#6 on most machines, or the thickness of two playing cards). The completely rolled sheet should measure 4–5" wide and 22–24" long.

Cut into three 8" sheets and place on floured kitchen towels. Repeat with remaining 3 pieces of dough. Turn sheets on kitchen towels occasionally so they dry slightly. They should be pliable but a bit leathery.

Bring a large pot of salted water to a boil. Working in 2 batches, cook pasta just until tender and color changes from deep yellow to pale yellow, about 1 minute. Drain and rinse briefly with cold water. Arrange lasagna sheets on clean kitchen towels, pressing them slightly to flatten. Use immediately or keep covered at room temperature up to 1 hour.