Heat a saucepan over medium-high heat. Add 2 teaspoons oil; swirl to coat. Add onion, carrot, and garlic; sauté 3 minutes. Add 3/4 cup water and quinoa; bring to a boil. Cover, reduce heat to medium, and simmer 18 minutes or until liquid is absorbed. Spoon cooked quinoa onto a large plate; cool 5 minutes. Combine quinoa mixture, panko, salt, pepper, chickpeas, and egg in a food processor; pulse until chickpeas are slightly mashed and mixture comes together. Shape quinoa mixture into 4 (1/2-inch-thick) patties.
Heat a large nonstick skillet over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Add patties to pan; cook 3 minutes on each side or until lightly browned.
Combine remaining 2 teaspoons oil and hot sauce in a bowl. Place patties on bottom halves of buns; top evenly with hot sauce mixture, lettuce, tomato, and top halves of buns.