Pat the chicken pieces dry and season lightly with black pepper. Dry surfaces char; wet ones steam.
Whisk together the gochugaru, gochujang, soy sauce, rice syrup, brown sugar, garlic, ginger and rice wine. Taste this raw sauce on the tip of a spoon and adjust the chili ratio now — it's the last easy moment to change the heat level.
Toss the chicken in about two-thirds of the sauce and let it sit 20 to 30 minutes at room temperature. Don't go longer than an hour; the sugar starts breaking the texture down.
Heat 1 tablespoon of oil in a wide pan or cast-iron skillet over high heat until it's shimmering, close to smoking.
Sear the chicken in a single layer, sauce and all, without stirring for the first 2 minutes. Let it catch color and char at the edges before you touch it.
Turn the pieces, reduce heat to medium, and add the remaining reserved sauce. Cook, stirring every minute or so, until the chicken is cooked through and the sauce has reduced to a glaze that coats the back of a spoon, about 6 to 8 minutes.
Stir in the minced fresh chili in the last minute of cooking, not before. Adding it late keeps the heat bright and sharp instead of letting it flatten into the background.
Take the pan off the heat, drizzle with sesame oil, and scatter sesame seeds and scallions over the top.
For cheese buldak: transfer the finished chicken to an oven-safe dish, cover with mozzarella, and broil 3 to 4 minutes until bubbling and browned in spots.