Empty the two 28-oz cans of whole peeled tomatoes into a large bowl and crush them with your hands until the pieces are about bite-size.
In a large saucepan over medium-high heat, warm 3 tablespoons extra-virgin olive oil. Add the 4 thinly sliced garlic cloves and cook, stirring, until the garlic begins to sizzle, about 1 minute.
Add the crushed tomatoes and 1 teaspoon kosher salt to the pan. Bring to a boil, then lower the heat and simmer, stirring occasionally, until the sauce is slightly reduced, about 30 minutes.
Remove the pan from the heat and whisk in 1 cup crème fraîche. Taste and season with additional kosher salt as needed. Set the sauce aside to cool to room temperature while you prepare to assemble the lasagna.
Preheat the oven to 400°F (205°C) and set an oven rack in the middle position. Have a 9 x 13-inch (23 x 33-cm) baking dish ready.
Spoon a thin layer of the room-temperature sauce onto the bottom of the baking dish and spread it to cover the surface.
Assemble the lasagna in layers using 3 no‑boil lasagna noodles per layer: lay 3 noodles in a single layer over the sauce (break noodles as needed to fit). Spoon over enough tomato sauce to cover the noodles. Sprinkle with a portion of the 1 cup finely grated Parmesan, a portion of the 1½ cups coarsely grated whole‑milk mozzarella, and a handful of the torn basil leaves. Divide the cheeses and basil roughly evenly among the layers.
Repeat step 7 three more times so you use all 12 noodles and all of the sauce, cheeses, and basil, finishing with sauce and cheese on top (do not leave the top layer as naked pasta or plain basil).
Bake the assembled lasagna, uncovered, until the top is nicely browned and the edges are bubbling, 35 to 40 minutes.
Remove from the oven and let the lasagna rest at room temperature for 15 minutes before slicing and serving.