Heat grill to high. In a medium bowl, use a fork to gently combine ground turkey with Gruyere, scallions, breadcrumbs, mustard, and garlic; season generously with salt and pepper. Gently form mixture into four 1-inch-thick patties.
Lightly oil grill. Place patties on hottest part of grill; sear until browned, 1 to 2 minutes per side. Move patties to cooler part of grill; continue grilling until cooked through, 5 to 10 minutes per side.