Place the two cans of whole peeled tomatoes in a large bowl and crush them by hand until they are in bite-size pieces; set aside.
Heat the olive oil in a large saucepan over medium-high heat. Add the sliced garlic and cook, stirring, until it begins to sizzle, about 1 minute.
Add the crushed tomatoes and 1 teaspoon kosher salt, bring to a boil, then reduce the heat and simmer, stirring occasionally, until slightly reduced, about 30 minutes.
Whisk in the crème fraîche until incorporated and taste for seasoning, adjusting salt as needed. Remove sauce from the heat and let cool to room temperature.
Preheat the oven to 400°F (205°C) and position a rack in the middle of the oven.
Spoon a thin, even layer of the room-temperature sauce into the bottom of a 9 x 13-inch baking dish to coat the surface.
Lay 3 no-boil lasagna noodles in a single layer over the sauce. Spread enough sauce to cover the noodles, then sprinkle with some Parmesan, mozzarella, and a portion of the torn basil.
Repeat the layering (noodles, sauce, Parmesan, mozzarella, basil) until ingredients are used, ending with sauce and cheese on top so the pasta and basil are not exposed.
Bake the lasagna uncovered until the top is browned and the edges are bubbling, 35 to 40 minutes.
Let the lasagna rest at room temperature for 15 minutes before slicing and serving to allow the pasta to absorb the sauce.
