For Domenica's sauce: Lightly saute the garlic and chili flakes in the olive oil in a saucepan. Add the tomatoes and salt and simmer for 20 to 30 minutes.
For the three-cheese filling: Mix together ricotta, goat cheese, Parmesan, basil, mint, nutmeg and some salt and pepper in a bowl.
For the pasta: Place the yolks in the base of a stand mixer fitted with the paddle attachment and turn on low speed. Start slowly adding flour. As the dough starts to come together, scrape down the sides of the bowl. Remove the dough and knead until smooth and not sticky. Let rest for 20 to 30 minutes.
Cut dough into four quarters. Roll the quarters through a pasta machine, starting with the widest opening, taking it through each opening once or twice and finishing with the smallest opening. (You should be able to see through the finished pasta sheets.)
Cut sheets into pieces that will fit in a 9-by-13-inch baking pan.
Boil a pot of water and cook pasta sheets for about 1 minute, then remove them and put them in ice water to stop the cooking. Remove the sheets and lay out on clean kitchen towels or napkins.
For the thyme bechamel: Heat the butter and flour in a saucepan over medium heat and stir together for a minute. Add the milk, thyme and some salt and pepper and cook until mixture starts to thicken. Turn off heat and set aside.
To assemble: Preheat oven to 375 degrees F.
Coat a large baking dish with olive oil. Lay a sheet of pasta in the bottom and spread with a thin layer of bechamel and cheese filling. Spread with a rubber spatula or large metal spoon. Add a layer of pasta, then spread a layer of tomato sauce on that.
Repeat this assembly until the pasta is all layered. Finish the top layer with tomato sauce and bechamel and sprinkle with Parmesan and basil. Cover with foil and bake for 24 to 30 minutes. Remove foil and cook for an additional 10 minutes.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.