Light a grill or preheat a grill pan. Brush the onion with oil and grill until charred and softened, about 5 minutes. Brush the cut sides of the buns with oil and grill until toasted, about 30 seconds.
Form the ground turkey into four 1-inch-thick patties and brush with oil. Season with salt and pepper and grill over high heat, turning once, until nearly cooked through, about 10 minutes. Top with the Gouda, close the grill and cook until the cheese is melted and the burgers are cooked through, about 1 minute longer.
Meanwhile, in a small bowl, combine the mustards, horseradish and honey and spread on the bottom half of the buns. Top with the burgers and onions. Close the burgers and serve right away.