Light a fire in a charcoal or gas grill.
In a medium bowl, combine the turkey meat, bread crumbs, lemon juice, lemon zest, thyme, salt, and pepper.
Divide the mixture into 12 equal portions and press them into 4-inch patties.
Top each of 6 patties with 1 tablespoon goat cheese.
Place a second patty on top and press down the edges to seal the two together.
Place the patties on a baking sheet, cover, and refrigerate for 1 hour.
Brush each burger with oil.
Grill on an oiled grill rack over a hot fire for 5 to 6 minutes on each side, or until cooked through.
Serve on an English muffin, topped with relish.