In a cast iron skillet or nonstick skillet, melt butter and heat up olive oil over medium high heat.
Season chicken with salt and pepper then place into hot skillet and brown on both sides, about 3-4 minutes on each side. You don't have to cook all the way through as you will do that in a later step.
Remove from skillet and place on a plate, set aside.
Add the sliced onions and mushrooms to the skillet and cook until onions are translucent and softened and mushrooms are softened and tender, about 5-7 minutes.
Use about 3-4 sprigs of thyme and pull the leaves off the stem and add to the skillet. Stir to incorporate.
Add the riesling to the skillet and scrape off the brown bits on the bottom of the skillet and let the onions and mushroom simmer for about 2-3 minutes in the wine.
Add the heavy cream and chicken stock, stir then add the remaining thyme (you may keep some on the stem like you see in the photos). Season with more salt and pepper.
Nestle the chicken into the mixture then bring the mixture to a rolling simmer and let simmer for about 10 minutes or until liquid has thickened and chicken as cooked through.