Prepare the tomato confit by combining the ingredients and baking in a 250°F (120°C) oven for 2-3 hours, or until the tomatoes are soft and jammy. Remove from the oven and set aside until ready to use.

Prepare the aioli by combining the ingredients well. Store in the refrigerator until ready to use.

Heat 1 tablespoon olive oil in a medium skillet, and sauté the mushrooms and shallots until well-cooked, 4-5 minutes. Set aside to cool.

In a medium bowl, combine the farro, panko, parsley, garlic powder, mayonnaise, egg, salt, and pepper until well-mixed. Add the cooled mushrooms and shallots.

Using your hands, form the burgers into four patties (if you find that they are not sticking together well, you may add a little more mayonnaise or beaten egg).

Heat the remaining tablespoon olive oil in a skillet over medium heat (you can use the same one in which you made the shallots and mushrooms) and add the shaped burger patties, cooking 3-4 minutes on each side until browned. Add a slice of cheddar atop each during the last 2 minutes of cooking, and cover.

Toast the buns by briefly heating in a skillet until browned (1-2 minutes). Transfer to a serving dish.

Divide the aioli between the buns, and add a burger patty. Top each patty with arugula and a few cherry tomatoes, and serve (you may discard or eat the whole garlic clove from the tomatoes).