In a medium bowl or container, dissolve 2 tablespoons kosher salt and 1 tablespoon brown sugar in 4 cups cold water. Add smashed garlic, bay leaf, peppercorns and thyme. Submerge the picantón in the brine, cover and refrigerate for 30 minutes to 2 hours. (If short on time, skip brining and proceed to seasoning.)

Rinse and dry

Remove bird from brine, discard brine, rinse the cavity briefly under cold water, and pat the bird thoroughly dry inside and out with paper towels. Let rest at cool room temperature while you prepare the filling (about 15 minutes).

Filling

Cook aromatics and meat

Heat a small skillet over medium heat and add 1 tablespoon olive oil. Add diced onion and red pepper and cook 3–4 minutes until softened. Add minced garlic and cook 30 seconds. Add ground pork (and crumbled chorizo, if using) and cook, breaking up with a spoon, until no longer pink, about 4–5 minutes. Drain any excess fat.

Combine filling

Transfer the cooked mixture to a bowl and let cool slightly. Stir in cooked rice, 1 tablespoon chopped parsley, 1 beaten egg, 1/2 teaspoon salt, 1/4 teaspoon black pepper, 1/2 teaspoon paprika and 1/4 teaspoon cumin. Mix until combined and moist but not loose. Taste and adjust seasoning.

Assemble & Roast

Preheat oven

Preheat oven to 425°F (220°C). Position a rack in the center of the oven. If you brined the bird, reduce oven to 400°F (200°C) after patting dry to avoid excessive browning; otherwise use 425°F.

Stuff the bird

Loosely spoon the filling into the cavity of the picantón—do not overpack; just enough so it holds shape. Tuck the legs together and secure with kitchen twine, and truss the wings under the body to prevent burning. Any remaining filling can be baked in a small ramekin alongside the bird.

Season and oil

Rub the outside of the bird with 1 tablespoon olive oil, then season evenly with 1 1/2 teaspoons kosher salt and 1/2 teaspoon freshly ground black pepper. If desired, sprinkle an extra 1/4 teaspoon smoked paprika over the skin for color.

Roast

Place the picantón breast-side up on a rack in a shallow roasting pan. Roast at 425°F/400°F according to whether you brined, for about 25–35 minutes for a 1–1.25 lb bird, or until an instant-read thermometer inserted into the thickest part of the thigh (avoiding bone) reads 165°F (74°C). If the skin browns too quickly, tent loosely with foil for the last 10 minutes.

Glaze and rest

During the last 5 minutes of roasting, brush the bird with a glaze made from 2 tablespoons melted butter mixed with 1 teaspoon honey and 1/4 teaspoon smoked paprika. Remove the bird from oven and transfer to a cutting board. Tent with foil and let rest 10 minutes to allow juices to redistribute.

Finish & Serve

Remove kitchen twine and carve the picantón. Serve warm with lemon wedges and sprinkle with remaining chopped parsley. Spoon any juices from the resting board over slices. If you baked the extra filling, serve alongside the carved bird.