In a small saucepan, combine the tomatoes, vinegar, sugar, allspice, cinnamon, ½ teaspoon salt, and ¼ teaspoon pepper.
Simmer, stirring occasionally, until the mixture thickens, 8 to 10 minutes. Set aside. (The chutney can be served warm or at room temperature.)
Heat grill to medium-high; oil the grate. In a medium bowl, combine the turkey, Gruyère, and garlic. (Do not overwork the meat.)
Form the mixture into four ¾-inch-thick patties. Make a shallow well in the top of each (to prevent overplumping while cooking). Season with the chili powder, ½ teaspoon salt, and ¼ teaspoon pepper.
Grill the patties, turning once (do not press or flatten), until an instant-read thermometer inserted in the center registers 160° F, 6 to 8 minutes per side.
Serve the burgers on the focaccia, topped with the chutney.