Marinate the bulgogi: In a large bowl, whisk together the brown sugar, soy sauce, wine, sesame oil, green onions, garlic, and pepper until well combined. Add the beef and coat it completely in marinade. Cover and refrigerate 4 to 5 hours.
To make the cucumber kimchi salad: In a medium bowl, combine the cucumbers, green onions, garlic, gochugaru, sugar, vinegar, sesame oil, and salt to taste to stir gently. Cover and refrigerate until ready to serve.
Prepare a hot grill. If the pieces of beef are so small that they may fall through the grates, use a grilling skillet or place a sheet of foil on the grill.
Grill the beef on both sides until medium-well, 3 to 5 minutes, flipping halfway through cooking. Don’t crowd the skillet or foil, so do this in batches if necessary. As you finish each batch, transfer it to a serving platter and continue with the remaining beef.
Serve the bulgogi on top of steamed rice. Garnish with green onion and toasted sesame seeds and spoon the cucumber kimchi salad alongside.
Store the leftover bulgogi and cucumber kimchi salad in separate covered containers in the refrigerator for up to 3 days.