Heat oven to broil.
Combine turkey, syrup, sage, cayenne, salt, pepper, and ginger in a mixing bowl and mix well. Divide into 4 (1/2-inch-thick) patties.
Heat oil in a large nonstick skillet over medium-high heat; cook patties until well browned and cooked through, about 4 minutes per side. Remove from heat.
Place tomato slices in a single layer in a baking dish; season with salt and pepper. Top each slice with 1 turkey patty and 1 cheese slice; broil until cheese is melted, about 2 minutes. Serve hot.