Pull the drumsticks from the refrigerator 20 to 30 minutes before cooking so they are not ice-cold in the center. Pat them completely dry with paper towels. Using a sharp knife, score the thickest part of the meat 2 to 3 times cutting about 1/2 inch deep. This helps heat and seasoning reach the center.

Preheat your oven to 375 degrees F. In a small bowl, stir together the salt, pepper, garlic powder, and paprika. Rub the seasoning all over the drumsticks, pressing it into the scored cuts and under any loose skin.

Heat the olive oil in a large oven-safe skillet over medium-high heat until the oil shimmers and you can feel the heat when you hold your hand a few inches above the pan. Add the drumsticks and sear without moving them for 3 to 4 minutes until the skin is deep golden brown and releases easily from the pan. Flip and sear the other side for another 3 to 4 minutes. Add the smashed garlic cloves to the pan around the drumsticks.

Transfer the entire skillet to the preheated oven. Roast uncovered for 40 to 45 minutes, turning the drumsticks once halfway through. The meat is done when a thermometer inserted into the thickest part away from the bone reads 175 degrees F and the juices run clear. The skin should be crispy and a rich mahogany color.

Carefully remove the skillet from the oven using heavy oven mitts. Transfer the drumsticks to a plate and tent loosely with foil while you make the sauce. Leave the garlic cloves and all the drippings in the pan.

Set the skillet over medium heat on the stovetop. Pour in the white wine and use a wooden spoon to scrape up all the browned bits stuck to the bottom of the pan. Those bits are pure flavor. Let the wine bubble and reduce for about 1 minute, then pour in the chicken broth and thyme. Simmer for 4 to 5 minutes until the sauce reduces by about one-third and thickens slightly.

Remove the pan from heat and swirl in the cold butter pieces one at a time until each one melts into the sauce. Taste and adjust salt if needed. Pour the pan sauce through a fine strainer or serve it with the garlic cloves smashed into it for extra richness. Spoon generously over the drumsticks and serve right away.