Preheat the oven to 400 degrees. In a skillet, cook the mushrooms in 2 tbsp. EVOO over medium heat, stirring, until golden, about 15 minutes. Toss in a bowl with 1 1/2 cups cheese; season.
In the skillet, cook the onion in the remaining EVOO over medium heat, stirring, until golden, 6 to 8 minutes. Add spices; cook for 2 minutes. Stir in tomato sauce and 1/2 cup water. Reduce heat; simmer for 5 minutes. Season with salt.
In a greased 9-by-13-inch baking dish, spread a third of the sauce. Coat tortillas in some sauce. Spread 1/3 cup mushroom filling down the centers; roll up and transfer to dish. Top with remaining sauce and cheese. Cover with foil and bake until bubbly, 15 minutes. Serve with rice and beans.