Make sauce by whisking together soy sauce, 6 Tbsp water, brown sugar, ketchup, Gochujang, rice vinegar, and toasted sesame oil. Set aside.
Heat a wok or large skillet over medium-high heat. Add cooking oil and when it begins to shimmer, add ground chicken. Saute, breaking the chicken apart, until chicken is nearly cooked through, 5 to 6 minutes.
Pour sauce over chicken and cook, stirring, until sauce begins to bubble.
In a small bowl, whisk together 2 Tbsp water and cornstarch to make a slurry.
Pour slurry over chicken and stir. Continue cooking until sauce thickens slightly, about 2 minutes more. Top chicken with sesame seeds.
Set out chicken, lettuce leaves, cooked rice (if using), roasted peanuts, and extra Gochujang. Assemble lettuce cups at the table and enjoy!