In a large bowl, crush the tomatoes with your hands (this is a messy but fun job—it’s a very good one for children) until they are in bite-size pieces.

In a large saucepan over medium-high heat, warm the olive oil, add the garlic, and cook, stirring, until it begins to sizzle, about 1 minute. Add the tomatoes and 1 tsp salt and bring to a boil. Lower the heat and let the sauce simmer, stirring every so often, until it is slightly reduced, about 30 minutes.

Whisk the crème fraîche into the sauce and season to taste with salt. Set the sauce aside to cool to room temperature while you conquer the pasta.

Preheat your oven to 400°F.

Ladle a thin layer of room-temperature sauce onto the bottom of a 9-by-13-in baking dish. Spread the sauce with a spoon to cover the surface of the dish. Add a layer of pasta (brush off any excess flour), cutting the pasta and arranging it as needed to form an even single layer. Spoon over just enough tomato sauce to cover the pasta and then scatter over some of the Parmesan, mozzarella, and basil. Repeat the layering process until you’ve used up all of your components, ending with sauce and cheese (not naked pasta or basil, both of which would burn if exposed).

Bake the lasagna, uncovered, until it’s gorgeously browned and the edges are bubbling, 35 to 40 minutes. Let it rest at room temperature for 15 minutes, just like you would a steak, before slicing and serving. This lets the pasta fully absorb all of the bubbling sauce, so you don’t end up with soupy slices.