Light charcoal briquettes in a grill and brush the grilling rack with oil to keep the salmon from sticking.
While the grill is heating, lay the salmon skin side down on a cutting board and cut it crosswise into 4 equal pieces. Whisk together the mustard, soy sauce, olive oil, and garlic in a small bowl. Drizzle half of the marinade onto the salmon and allow it to sit for 10 minutes.
Place the salmon skin side down on the hot grill; discard the marinade the fish was sitting in. Grill for 5 minutes, depending on the thickness of the fish. Turn carefully with a wide spatula and grill for another 5 minutes. The salmon will be slightly raw in the center, but don't worry; it will keep cooking as it sits.
Transfer the fish to a flat plate, skin side down, and spoon the reserved marinade on top. Allow the fish to rest for 10 minutes. Remove the skin and serve warm.