Crush the canned tomatoes by hand in a large bowl until broken into bite-size pieces.
In a large saucepan over medium-high heat, warm the olive oil. 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 heat and simmer, stirring occasionally, until slightly reduced, about 30 minutes.
Whisk the crème fraîche into the sauce and taste; add more salt if needed. Remove the sauce from the heat and let it cool to room temperature.
Preheat the oven to 400°F (205°C) and position a rack in the middle.
Spoon a thin layer of room-temperature sauce into the bottom of a 9 x 13-inch baking dish and spread to coat the surface.
Arrange 3 no-boil lasagna noodles in a single layer over the sauce. Spoon just enough sauce to cover the noodles, then sprinkle with some Parmesan, some mozzarella, and scatter a portion of the torn basil.
Repeat layering with noodles, sauce, Parmesan, mozzarella, and basil until all components are used, ending with sauce and cheese on top.
Bake the lasagna, uncovered, until the top is nicely browned and edges are bubbling, 35 to 40 minutes.
Let the lasagna rest at room temperature for 15 minutes before slicing and serving so the pasta absorbs the sauce and slices hold together.