1. Preheat the broiler to medium-high with the rack 4 inches from the heat source. Grease a rimmed baking sheet with the oil.
2. Combine the garlic and one-third of the salmon in a food processor. Process to a purée, stopping to scrape down the sides of the container as necessary.
3. Add the remaining salmon along with the onion, bell pepper, parsley, and a pinch each of salt and pepper. Pulse until the salmon is chopped, but not too finely. Shape the mixture into 4 patties and put on the prepared baking sheet.
4. Broil, undisturbed, until the side exposed to the heat develops a browned crust (3-5 minutes). Carefully turn the burgers and repeat on the other side, cooking until the salmon has turned opaque throughout (3-5 minutes more). Serve on buns with the basil, tomato, relish, and yogurt.