Prepare the ingredients & make the black bean mayo:

If you’re grilling, preheat your grill to maintain a temperature of 450-500°F. Oil the grill grates. Wash and dry the fresh produce. Cut out and discard the core of the cabbage; thinly slice the leaves. Cut off and discard the stems of the peppers; remove and discard the cores. Halve lengthwise, then thinly slice crosswise. Thinly slice the cucumber into rounds. If necessary, peel the garlic, then finely chop. Using the flat side of your knife, smash until it resembles a paste (or use a zester). Thinly slice the cheese. Halve the buns. In a bowl, combine the mayonnaise, black bean sauce, and a drizzle of olive oil. Season with salt and pepper to taste.

Make the slaw:

In a bowl, combine the sliced cabbage and peppers, half the vinegar, 1 teaspoon of olive oil, and as much of the gochujang as you’d like, depending on how spicy you’d like the dish to be. Season with salt and pepper. Set aside to marinate, stirring occasionally, at least 10 minutes. Season with salt and pepper to taste.

Marinate the cucumber:

While the slaw marinates, in a bowl, combine the sliced cucumber, remaining vinegar, and up to half the garlic paste. Drizzle with olive oil and season with salt and pepper. Set aside to marinate, stirring occasionally, at least 10 minutes. Season with salt and pepper to taste.

Form & cook the patties:

In a bowl, combine the beef, half the breadcrumbs (you will have extra), and remaining garlic paste; season with salt and pepper. Gently mix to combine. Using your hands, form the mixture into two 1/2-inch-thick patties; transfer to a plate. 

STOVE: In a medium pan (nonstick, if you have one), heat a drizzle of olive oil on medium-high until hot. Add the patties and cook 4 to 5 minutes on the first side, or until browned. Flip the patties (carefully, as the oil may splatter). Evenly top with the sliced cheese. Loosely cover the pan with aluminum foil and cook 4 to 5 minutes, or until the cheese is melted and the patties are cooked through. Transfer to a plate. Wipe out the pan.
GRILL: Lightly oil the patties on both sides. Grill 6 to 7 minutes on   the first side, or until browned. Flip and continue to grill 4 minutes. Evenly top with the sliced cheese. Continue to grill 2 to 3 minutes, or until the cheese is melted and the patties are cooked through. Transfer to a plate.

Toast the buns & serve your dish:

STOVE: Heat the same, dry pan on medium-high until hot. Add the buns, cut side down, and toast 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface.
GRILL: Grill the buns, cut side down, 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface. 

Assemble the burgers using the toasted buns, black bean mayo, cooked patties, and marinated cucumber (discarding any liquid). Serve with the slaw on the side. Enjoy!