- Rinse and drain 3 pounds trimmed chicken drumsticks. Transfer to large bowl. Add 2½ cups milk and toss to coat. Arrange chicken in even layer, making sure all chicken is submerged in milk. Cover and refrigerate for 1 hour.
- Drain and thoroughly rinse chicken until water runs clear. Clean and dry bowl.
- Using sharp knife, make 4 to 6 slashes in each drumstick (each slash should reach bone). Pat chicken dry with paper towels and transfer back to bowl.
- Add ¼ cup extra-virgin olive oil, ½ teaspoon fine salt, and ½ teaspoon pepper, and toss chicken to coat. Cover and refrigerate for at least 1 hour or up to overnight.
- Meanwhile, whisk ¼ cup honey, 2 tablespoons soy sauce, 2 tablespoons distilled white vinegar, 2 tablespoons sugar, ½ teaspoon salt, and ½ teaspoon pepper together in small bowl. Set sauce aside.
- Remove chicken from fridge and toss again to coat in oil that has settled in bowl. Working with 1 drumstick at a time, stretch skin to evenly cover meat, then arrange skin-side down in 12‑inch nonstick skillet. Drizzle any remaining oil left in bowl over chicken.
- Cook over medium-high heat, turning chicken occasionally, until spotty golden brown, about 10 minutes.
- Reduce heat to medium, cover, and continue to cook, turning chicken occasionally, until it registers at least 185 degrees, about 20 minutes.
- Spoon 6 tablespoons finely chopped garlic into spaces between drumsticks where oil has pooled. Cook, spreading garlic into skillet with back of a spoon, until fragrant, about 30 seconds. Gently toss chicken until evenly coated with garlic.
- Add reserved sauce, stirring in any browned bits in skillet. Increase heat to medium-high and cook, continuously coating and basting chicken in sauce, until sauce has thickened, about 2 minutes.
- Stir in butter until sauce has thickened into glaze, about 30 seconds. Serve immediately.
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