Place the cooked lentils and black beans in a large bowl. Mash with a fork or potato masher until mostly combined but still slightly chunky (about 1–2 minutes).

Add dry binders and flavor

Add the rolled oats, grated carrot, finely chopped onion, minced garlic, smoked paprika, ground cumin, soy sauce, chopped parsley, salt, and pepper to the bean-lentil mix. Stir well until evenly distributed. Let the mixture sit for 5 minutes so the oats absorb moisture and tighten the mixture.

Form patties

Divide the mixture into 4 equal portions and shape each into a patty about 3/4-inch thick and slightly larger than the bun diameter. If mixture is too wet, add up to 1–2 tbs more oats; if too dry, add 1–2 tbs water.

Cook patties

Heat 2 tbs neutral oil in a large nonstick or cast-iron skillet over medium heat. When hot, add patties (work in batches if needed) and cook 4–5 minutes per side until a golden-brown crust forms. Press gently once with a spatula to ensure even contact. Reduce heat if browning too quickly.

Melt cheese

During the last minute of cooking, place a slice of cheddar on each patty and cover the pan briefly (30–60 seconds) to melt the cheese. Remove patties and keep warm on a plate tented with foil.

Coleslaw

In a medium bowl combine mayonnaise, yogurt (or sour cream), apple cider vinegar, honey (or maple syrup), salt, and pepper. Whisk until smooth.

Add shredded cabbage and grated/julienned carrot to the dressing. Toss thoroughly to coat. Taste and adjust seasoning with additional vinegar, salt, or sweetener as desired. Refrigerate until assembly (at least 10 minutes) so flavors meld.

Assembly & extras

Toast buns: Heat a clean skillet over medium. Spread softened butter on cut sides of brioche buns and toast buttered sides down until golden, about 1–2 minutes.

Assemble burgers: On the bottom half of each toasted bun place a lettuce leaf, a cheese-topped patty, 1–2 tomato slices, and about 2–3 tbs coleslaw. Add any optional sauce to the top bun if desired. Close with the top bun.

Serve: Serve immediately while patties are warm and cheese is melty. Recommended sides: fries, potato wedges, or a simple green salad.