Rach's Easy Skillet Lasagna with Mushrooms

Name: Rach's Easy Skillet Lasagna with Mushrooms
Cuisine: Italian
Ingredients: 1 pound flat pasta squares or sheets; dried; Salt; 3 cups vegetable stock or water or combo; 1 cup dried porcini; 3 tablespoons EVOO; 1 onion; finely chopped; 1 large carrot; peeled and finely diced; 1 ¼ to 1 ½ pounds fresh mixed mushrooms; cleaned and sliced or pulled into thin pieces; Pepper; 2 tablespoons fresh thyme; chopped; 2 cloves garlic; grated; ¼ cup tomato paste; 1 cup white wine; 2 cups diced fresh peeled tomatoes or chopped canned tomatoes (28 ounces); 1 cup heavy cream; 1 cup torn basil leaves; 2 cups grated cow's and sheep's milk cheese (pecorino and Parm) combined; Fresh parsley; stemmed and chopped