Preheat the oven to 400°F (200°C).
Pat the chicken leg quarters dry with paper towels.
In a small bowl, mix together salt, black pepper, and paprika.
Rub the seasoning mixture evenly over the chicken leg quarters.
Heat olive oil in a large oven-safe skillet over medium-high heat.
Add the chicken leg quarters skin-side down and sear for 5-6 minutes until the skin is golden brown.
Flip the chicken and cook for an additional 3 minutes.
Transfer the skillet to the preheated oven and roast the chicken for 25-30 minutes or until the internal temperature reaches 165°F (74°C).
Remove the chicken from the skillet and set aside, keeping it warm.
Place the skillet back on the stove over medium heat (if your skillet is not oven-safe, transfer the drippings to a saucepan).
Add the butter to the skillet and melt.
Add the chopped onion and sauté for 3-4 minutes until translucent.
Add the minced garlic and cook for 1 minute until fragrant.
Add the sliced mushrooms and cook for 6-8 minutes until they release their moisture and begin to brown.
Sprinkle the flour over the mushrooms and stir well to coat. Cook for 1-2 minutes to remove the raw flour taste.
Gradually pour in the chicken broth while stirring constantly to prevent lumps.
Add the fresh thyme leaves and dried rosemary.
Simmer the gravy for 5-7 minutes, stirring occasionally, until it thickens.
Season the mushroom gravy with salt and black pepper to taste.
Serve the chicken leg quarters topped with the mushroom gravy.