Cheesecake Factory Pasta with Mushroom Bolognese

6 ounces spaghettini, cooked al dente, drained and kept warm
2 ounces olive oil
1 ounce carrots, minced
1 ounce yellow onion, diced to 1/4-inch thickness
4 ounces mushrooms, diced to 1/4-inch thickness
1/2 teaspoon kosher salt
1/4 teaspoon black pepper, freshly ground
1 tablespoon garlic, minced
1 teaspoon fresh thyme, chopped
2 ounces Madeira wine
10 ounces marinara sauce
1 ounce butter
1 ounce Parmesan cheese, grated
2 teaspoons parsley, chopped

In a saut