Heat 1 tablespoon butter and 1 tablespoon of the olive oil in a large frying pan over medium heat. Add the shallots and cook until transparent and lightly golden around edges, about 5-6 minutes, stirring every couple minutes. Add the garlic, thinly sliced mushrooms, salt, pepper, thyme, and remaining 1 tablespoon olive oil and stir to combine. Cook until mushrooms are more flexible and have released their moisture and are fragrant, about 10-15 minutes more, stirring every 2-3 minutes.

Begin preparing the gnocchi according to the package directions (use salted water to add flavor to the gnocchi). Meanwhile, you can finish the mushroom sauce. Sprinkle the flour over the mushroom mixture and stir to coat, then add the wine and stir to combine, dissolving any browned bits on the bottom of the frying pan. Cook for one minute more, then add the vegetable stock and cream and stir until a thick sauce forms. Stir in the grated pecorino romano and the fresh basil leaves and remove from heat. Toss with the cooked and drained gnocchi and serve immediately.