Preheat the oven to 425°F (220°C).
Pat the chicken leg quarters dry with paper towels.
In a small bowl, combine 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 4-5 minutes until the skin is golden brown and crispy. Flip 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 cover loosely with foil to keep warm.
Place the skillet back on the stove over medium heat. Add the butter and let it melt.
Add the sliced mushrooms and cook, stirring occasionally, until they release their moisture and begin to brown, about 6-8 minutes.
Add the chopped onion and cook until softened, about 3 minutes.
Stir in the minced garlic and cook for 30 seconds until fragrant.
Sprinkle the flour over the mushroom mixture and stir well to combine. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in the chicken broth, ensuring no lumps form.
Add the fresh thyme leaves and bring the gravy to a simmer. Cook, stirring occasionally, until the gravy thickens, about 5 minutes.
Stir in the heavy cream and season with salt and black pepper to taste. Cook for an additional 2 minutes.
Serve the chicken leg quarters topped with the mushroom gravy.