How to Make It
In a large shallow dish, mix together 1 tablespoon each of the basil and parsley, 1 teaspoon of the tarragon, 1 tablespoon of the olive oil and 1 minced garlic clove. Rub the herb mixture all over the lamb chops and season them generously with salt and pepper. Let the lamb chops stand at room temperature for at least 30 minutes and for up to 2 hours.
Meanwhile, preheat the oven to 425°. In a large skillet, bring the water to a boil. Add the carrots and a pinch of salt and cover. Simmer the carrots over low heat for 4 minutes. Add the turnip, cover and cook until both vegetables are tender, about 8 minutes. Drain the vegetables and transfer them to a plate. Return the skillet to low heat. When hot, add 2 tablespoons of the olive oil and the shiitake and season generously with salt and pepper. Cover and cook the shiitake, stirring occasionally, until lightly browned, about 4 minutes. Add the chanterelles, cover and cook until tender, about 3 minutes. Add the shallots and the remaining 2 garlic cloves and cook, stirring, until fragrant, about 3 minutes. Add the carrots and turnip, cover and remove the vegetables from the heat.
In a large ovenproof skillet, heat the remaining 2 tablespoons of olive oil until shimmering. Add the lamb chops and cook them over moderately high heat until browned on the bottom, about 3 minutes. Turn the lamb chops over and roast them in the oven for about 8 minutes for medium rare. Transfer the lamb chops to a plate, cover with foil and keep warm.
Discard the fat from the skillet and set it over moderately high heat. Add the vermouth and cook until almost evaporated, about 3 minutes. Add the chicken stock and boil until reduced to 3/4 cup, about 3 minutes. Add any accumulated lamb juices and pour the liquid over the reserved vegetables. Stir in the cherry tomatoes and simmer until hot. Stir in the remaining 1 tablespoon each of chopped basil and parsley and 1 teaspoon of chopped tarragon. Season the ragout with salt and pepper.
Spoon the vegetable ragout into shallow bowls and sprinkle with Parmesan cheese. Top each plate with a lamb chop and serve at once.