In a mixing bowl, whisk together the soy sauce, chicken broth, sweet chili sauce, Sriracha, brown sugar, lime juice, minced garlic, and grated ginger until smooth.
Season the drumsticks lightly with salt and black pepper, then place them in a large resealable bag and pour the marinade over the chicken. Seal the bag and refrigerate for a few hours or up to overnight.
Transfer the chicken and all marinade into a 6-qt slow cooker. Cover and cook on LOW for 4–5 hours (or on HIGH for 3–4 hours) until the chicken is cooked through and tender.
Carefully remove the drumsticks from the slow cooker and place them on a rimmed baking sheet, spacing them apart.
Set the oven broiler to high and broil the drumsticks until the skin is crispy and slightly charred, watching closely so they don’t burn.
Check with a meat thermometer to ensure the internal temperature reaches 165°F, then serve garnished as desired.