Lamb Chops with Vegetable Ragout

Name: Lamb Chops with Vegetable Ragout
Ingredients: 2 tablespoons coarsely chopped basil leaves; 2 tablespoons chopped flat-leaf parsley; 2 teaspoons chopped tarragon; 5 tablespoons extra-virgin olive oil; 3 garlic cloves, minced; 4 double lamb loin chops (8 to 10 ounces each); Salt and freshly ground pepper; 2 cups water; 2 medium carrots, cut into 2-inch matchsticks; Salt; 1 large turnip, peeled and cut into 2-inch matchsticks; 1/2 pound shiitake mushrooms, stems discarded, caps quartered; 1/4 pound small chanterelles, trimmed; 3 tablespoons very finely chopped shallots; 1/2 cup dry French vermouth; 1 cup chicken stock or canned low-sodium broth; 8 cherry tomatoes, halved; Freshly grated Parmesan cheese, for serving