To make the quinoa: In a small pot, bring the water to a simmer over high heat, then add the quinoa and let the water return to a simmer. Once simmering, turn the heat to low, cover the pot with a lid and let it cook until the water is absorbed, about 15 minutes.
To make the burgers: In a large bowl, smash the black beans with a fork until they have a paste-like texture. Leave some beans whole.
Add the breadcrumbs, cooked quinoa, yellow bell peppers, white onions, garlic, cumin, salt, hot sauce and egg to the bowl. Mix with your hands until fully incorporated.
Divide the mixture into 5 even portions and form them into patties.
In a large skillet, heat the olive oil over medium heat. Add the patties to the pan and cook until browned, 2 to 3 minutes per side.
Transfer the patties to the buns and garnish with the arugula, tomato and red onion.