In a food processor add the cooked black beans and pulse 6–8 times until mostly mashed but still a bit chunky. Add the rolled oats, grated carrot, chopped mushrooms, chopped red onion, minced garlic, ground cumin, smoked paprika, soy sauce, tomato paste, salt and pepper. Pulse in 6–10 short bursts until mixture holds together when pressed but still has texture. Scrape down sides as needed.

Combine and chill

Transfer mixture to a bowl, stir in the thickened flax egg until evenly combined. If mixture feels too wet add 1–2 tablespoons more oats; if too dry add 1 tablespoon water. Cover and refrigerate 15 minutes to firm up for easier shaping.

Form patties

Divide the mixture into 4 equal portions and shape each into a patty roughly 3–4 inches in diameter and about 3/4 inch thick. Press edges firmly so they hold together.

Cook patties

Heat a large nonstick or cast-iron skillet over medium heat and add 1 tablespoon olive oil. When shimmering, add two patties and cook 4–5 minutes per side until a browned crust forms and patties are heated through. Add the remaining 1 tablespoon oil and cook the other two patties the same way. Alternatively, bake at 200°C/400°F for 12–15 minutes, flipping halfway, until browned.

Sauce / Spread

While patties chill or cook, whisk together the Greek yogurt, mayonnaise, lemon juice, Dijon mustard, hot sauce (if using), salt and black pepper in a small bowl. Taste and adjust seasoning. Keep chilled until assembly.

Lettuce wraps & assembly

Prepare lettuce leaves by patting them dry with paper towels and trimming large stems so leaves lay flat. Arrange leaves on a platter.

Warm patties briefly if needed (30–60 seconds in microwave or return to hot skillet for 30 seconds) so they are warm for assembly.

Spread about 1 tablespoon of the sauce onto the center of each lettuce leaf. Place a veggie patty on top of the sauce. Top each patty with a slice of tomato, a few avocado slices, a sprinkle of sliced red onion and shredded carrot, and a bit of chopped cilantro or parsley.

Fold lettuce around fillings to form a wrap and serve immediately. Optionally serve extra sauce on the side for dipping.