Pat chicken pieces dry with paper towels, then season generously with salt and paprika. Heat olive oil in a large skillet over medium-high heat, and sear chicken for 4-5 minutes per side until golden brown—don't overcrowd the pan.
Remove chicken and set aside on a clean plate, keeping all the flavorful browned bits in the skillet. Reduce heat to medium, add 2 tbsp butter, and sauté diced onion until softened, approximately 3-4 minutes.
Add minced garlic and cook for 30 seconds until fragrant, then stir in sliced mushrooms and cook for 6-7 minutes until they release their moisture and begin to brown.
Sprinkle flour over the mushroom mixture and stir constantly for 1 minute to create a roux base. Pour in white wine and scrape up the browned bits from the skillet bottom, letting it simmer for 2 minutes until the wine reduces slightly.
Return the cozy creamy mushroom chicken recipe components to the skillet by adding chicken back to the pan along with chicken broth and dried thyme. Bring to a simmer, then reduce heat to low and cook covered for 20-25 minutes until chicken is cooked through and tender.
Remove from heat and stir in heavy cream and remaining 2 tbsp butter until the sauce becomes silky and smooth. Taste and adjust seasoning as needed, then garnish generously with fresh parsley before serving.