Heat 2 tablespoons of oil over medium-high heat in a large (12-inch) skillet. When oil is hot, add mushrooms; cook without stirring until golden brown on one side, about 5 minutes. Stir, seasoning with salt, pepper, and 1/2 teaspoon thyme. Continue to cook until impressively brown, a couple of minutes longer. Transfer to a bowl and set aside.
Return skillet to the burner and heat remaining 2 tablespoons of oil. Add onions and carrots; sauté until tender, about 5 minutes. Stir in remaining thyme, along with wine, followed by broth and then tomato paste, vinegar, mushrooms and beans. Bring to a simmer, reduce heat to medium-low, and simmer to blend flavors, about 10 minutes.
Whisk in enough of the cornstarch mixture to desired thickness. Adjust seasonings and serve over bowls of warm polenta, rice, or a mashed potatoes.