Preheat the oven to 425°F (220°C).
Pat the chicken leg quarters dry with paper towels.
In a small bowl, mix together kosher salt, black pepper, garlic powder, 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 and crispy.
Flip the chicken and sear the other side for 3 minutes.
Transfer the skillet to the preheated oven and roast the chicken for 30-35 minutes, or until the internal temperature reaches 165°F (74°C).
Remove the chicken from the skillet and set aside, tenting with foil to keep 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 sliced mushrooms and cook for 6-8 minutes until they release their moisture and begin to brown.
Stir in the minced garlic and cook for 1 minute until fragrant.
Sprinkle the flour over the mushroom mixture and stir well to combine. Cook for 2 minutes to remove the raw flour taste.
Gradually whisk in the chicken broth, making sure to scrape up any browned bits from the bottom of the skillet.
Add the fresh thyme leaves, salt, and black pepper.
Bring the gravy to a simmer and cook for 5-7 minutes until thickened.
Taste and adjust seasoning if needed.
Serve the chicken leg quarters hot with the mushroom gravy spooned over the top.