Combine quinoa, 1 cup water, and ½ tsp. salt in a small saucepan. Bring water to boil, cover pan and reduce heat to a low simmer.
Cook quinoa until tender, about 15 minutes. Set aside to cool slightly.
While quinoa is cooking, heat 1 teaspoon olive oil in a skillet over medium-high heat.
Add onion and red pepper to pan. Cook, stirring frequently, until browned and softened.
Reduce heat to medium and allow pan to cool slightly.
Stir garlic, paprika, cumin, oregano, red pepper flakes, and ½ tsp. salt into pan. Cook, stirring constantly, until fragrant (about 30 seconds). Remove from heat and allow to cool slightly.
Mash black beans into a paste. Combine black beans, flour, egg, quinoa, and onion/red pepper. Mix well.
Stir edamame and green onion into black bean mixture. Taste and add more salt, if necessary.
Shape mixture into 4 patties.
Heat remaining 2 teaspoons olive oil in a skillet over medium heat. Add burgers and cook until browned on one side, then flip.
Top burgers with cheddar cheese.
Continue cooking burgers until browned on bottom and cheese is melted.
To Serve:
While burgers are cooking, mix together mayonnaise, barbeque sauce, and garlic clove.
Serve burgers on buns with lettuce and tomato, topped with sauce. Enjoy!