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 4 to 5 minutes, depending on the thickness of the fish. Turn carefully with a wide spatula and grill for another 4 to 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, at room temperature, or chilled.