Bring a large pot of lightly salted water to a boil. Cook pasta shells in boiling water, stirring occasionally, until cooked through but firm to the bite, about 8 minutes; drain.
Heat olive oil in a large skillet over medium-high heat. Cook and stir mushrooms and garlic in hot oil until mushrooms are soft, about 7 minutes. Stir spinach and thyme into the mushroom mixture; season with salt and pepper. Cook and stir the mixture until the spinach wilts.
Remove the skillet from heat and drain any excess liquid; transfer drained mixture to a large bowl.
Preheat oven to 375 degrees F (190 degrees C).
Mix ricotta cheese, 3/4 cup mozzarella cheese, and 1/2 cup Parmesan cheese with the spinach mixture; season again with salt and pepper and stuff into pasta shells.
Arrange stuffed shells into a 9x11-inch baking dish.
Pour pasta sauce over the shells; top with 1/4 cup mozzarella cheese and 1/4 cup Parmesan cheese.
Bake in preheated oven until the cheese is melted and the shells are hot, about 30 minutes.
