Mix turkey, 1 teaspoon oil, salt, and pepper in medium bowl. Shape into two 1/2-inch-thick patties. Place 1 toasted bread slice on each of 2 plates; spread each with mayonnaise.
Heat 2 tablespoons oil in large nonstick skillet over medium-high heat. Add mushrooms, shallots, and garlic; sauté 2 minutes, then push to side of skillet. Add turkey patties to other side of skillet. Cook until brown on bottom, about 4 minutes. Turn patties over; top with cheese. Cook until patties are cooked through, cheese melts, and mushrooms are brown, stirring mushrooms often, about 4 minutes. Season mushrooms with salt and pepper.
Meanwhile, whisk vinegar and 2 teaspoons oil in small bowl; mix in arugula. Sprinkle with salt and pepper.
Spoon mushrooms onto bread; top with patties and arugula salad.
