Charred Vegetable Ragù

Name: Charred Vegetable Ragù
Ingredients: 4 large portobello mushrooms, stemmed and quartered; 1 garlic head; 1 yellow onion, peeled and quartered; 2 celery stalks, cut into thirds; 2 carrots, peeled and cut into thirds; 6 tablespoons extra-virgin olive oil, divided; 8 ounces fresh cremini mushrooms, stemmed and quartered; 1/2 cup tomato paste; 2 teaspoons dried oregano; 1/2 cup dry red wine; 1 teaspoon kosher salt, plus more to taste; 1/2 teaspoon black pepper; 12 ounces uncooked bronze-cut durum wheat tagliatelle pasta (such as Seggiano Organic); 2 ounces Parmigiano-Reggiano cheese, grated (about 1/2 cup), plus more for serving