Pulse mushrooms in a food processor until finely chopped, and transfer to a bowl. Shred zucchini, place on paper towels, and squeeze to remove excess moisture. Add to mushrooms.
Heat 2 tablespoons oil in a large pan over medium heat. Add shallot and red-pepper flakes, and cook until softened, about 2 minutes. Add mushrooms and zucchini, and cook until tender, about 2 minutes. Remove from heat, and add Parmesan, quinoa, and salt. Season with pepper. Let cool completely, then stir in egg and breadcrumbs. Cover, and refrigerate until cold and firm, about 1 hour.
Heat remaining 2 tablespoons oil in a large nonstick skillet over medium heat. Shape mixture into six 1/2-inch-thick patties, pressing firmly. Cook in batches until golden brown and cooked through, about 3 minutes per side.
Wipe pan clean, and return to medium heat. Brush cut sides of buns with oil, and heat buns, cut sides down, until toasted, about 1 minute. Spread yogurt-garlic sauce on cut sides of bun halves. Divide burgers, sprouts, and cucumber among bottom halves. Sandwich with top halves.
These patties are softer than a traditional veggie burger. The pan-searing helps the patty hold together and forms a crisp crust.