Put oil, garlic, and steak in a medium bowl and turn to coat. Let marinate 20 minutes at room temperature. Mix together butter and chives; set aside.
Prepare grill for high heat (450° to 550°). Grill meat, turning once, until grill marks appear and meat is cooked the way you like, 8 minutes for medium rare. Cover with foil and let rest 10 minutes, then slice thinly across the grain.
Spread one side of bread slices with butter mixture. Layer 4 slices with arugula, some sliced steak, and slices of radish. Top with remaining bread slices.