Wash and dry the fresh produce. Thinly slice the cucumber into rounds. In a bowl, combine half the sugar and half the vinegar; whisk until the sugar has dissolved. Add the sliced cucumber; toss to combine. Thinly slice the scallions, separating the white bottoms and hollow green tops. Peel the carrots and grate on the large side of a box grater. Cut out and discard the core of the cabbage; thinly slice the leaves. Combine the grated carrots and sliced cabbage in a bowl. Add the sliced green tops of the scallions, remaining vinegar, and half the sesame oil; season with salt and pepper. Set aside to marinate, stirring occasionally, at least 10 minutes. Taste, then season with salt and pepper if desired.

Meanwhile, in a bowl, combine the beef, sliced white bottoms of the scallions, soy sauce, remaining sugar, and remaining sesame oil. Season with salt and pepper; gently mix to combine. Form the mixture into two ½-inch-thick patties.  

In a medium pan (nonstick, if you have one), heat a drizzle of olive oil on medium-high until hot. Add the patties. Cook 4 to 5 minutes per side (flipping carefully, as the oil may splatter), or until browned and cooked through.* Transfer to a plate. Carefully drain off and discard any excess oil.

*An instant-read thermometer should register 160°F.

Meanwhile, halve the buns. Add the halved buns, cut side down, to the same pan. Toast on medium-high 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface.

In a bowl, combine the mayonnaise and as much of the gochujang as you’d like, depending on how spicy you’d like the dish to be. Assemble the burgers using the toasted buns, gochujang mayo, cooked patties, and dressed cucumber (discarding any liquid). Serve the burgers with the slaw on the side. Enjoy!