Place a soup pot over medium-high heat with 4 turns of the pan of EVOO, about 4 tablespoons. Add the mushrooms and garlic to the pan, and cook until tender, 5-6 minutes.
Add the barley, reserved veggies from the night before and beef stock, and bring the liquids to a bubble. Season with salt and pepper and simmer the mixture until the barley is tender, about 20 minutes.
When the barley has cooked through, add the leftover meat and sauce to the pan and reserve warm.
Place a large skillet over medium heat with the butter. When the butter has melted, add the sage leaves to the pan and fry until dark green and crispy, 2-3 minutes.
While the sage is frying, lay down four pieces of bread and top each with 2 slices of Fontina. Remove the sage leaves from the pan and lay 4 of them in each sandwich. Cap the sammies off with the remaining bread slices and add them to the pan that the sage was fried in. Cook until golden brown and the cheese has melted, about 5 minutes per side.
Serve the soup along with the Fontina-sage grilled cheese sammies.