Preheat the oven to 350°F (180°C) and line a baking sheet with parchment paper or foil.
Pat the chicken leg quarters dry with paper towels.
In a small bowl, whisk together the brown sugar, paprika, salt, dried thyme, garlic powder, onion powder, and black pepper.
Pour the olive oil over the spice mixture and stir to form a loose paste.
Place the chicken on the prepared baking sheet and rub the oil-spice mixture all over each piece, coating all surfaces.
Bake in the preheated oven for about 40 minutes, until the thickest part of the meat reaches 175°F (79°C).
Remove the chicken from the oven and let rest for 10 minutes before serving.