In a large nonreactive saucepan, bring satsuma juice, pepper jelly, rum, vinegar, ⅛ teaspoon salt, and bay leaves to a boil over medium-high heat. Boil until reduced to ¾ cup, 15 to 17 minutes. Remove from heat; discard bay leaves. Reserve ¼ cup satsuma mixture for serving.
Preheat oven to 450°.
In a large skillet, heat 1 tablespoon oil over medium-high heat. Sprinkle quail with pepper and remaining 1 teaspoon salt. Add half of quail to skillet; cook until browned, 1 to 2 minutes per side. Place on a large rimmed baking sheet. Repeat with remaining 1 tablespoon oil and remaining quail. Brush quail with satsuma mixture.
Bake for 7 minutes. Turn quail, and brush with satsuma mixture. Bake until an instant-read thermometer inserted in thickest portion registers 165°, about 7 minutes more, brushing with pan juices. Place on a serving platter. Sprinkle with thyme. Drizzle with reserved ¼ cup satsuma mixture. Garnish with satsuma slices and bay leaves, if desired.
Recipe by Louisiana Cookin' at https://louisianacookin.com/satsuma-lacquered-quail/