Preheat oven to 375 F. 

Heat a large cast iron skillet (or another heavy, oven safe skillet) over medium-high heat on the stove. Add butter and 1 tbsp canola oil. When butter is completely melted, add the onions, and use a wooden spoon to gently separate the pieces. Cook over medium heat, stirring occasionally. After about 5 minutes, add a pinch of salt, and continue to cook until brown, about 15 minutes. Remove onions and set aside in a bowl.

Lightly season the chicken pieces with salt and pepper. Add 2 tbsp canola oil to the skillet, and raise heat to medium-high. Add flour to a dish (I used a baking dish with raised sides), and gently coat each piece of chicken in the flour, shaking off the excess (use tongs to make this mess-free). Place about half of the coated pieces of chicken in the skillet, and cook on each side for about 4 minutes until browned. Remove the browned pieces to a new dish, and repeat the process with the remaining chicken pieces. 

When all of the chicken has been browned and set aside, deglaze the pan by adding the white wine, and use a wooden spoon to scrape up browned bits from the bottom. Cook for 1-2 minutes, then add all of the chicken pieces to the skillet (I had to really squish mine in there). Cook the chicken in the wine for about 2 minutes on each side. Top the chicken evenly with the browned onions, and place in the oven for 20-25 minutes until cooked through. 

If using cheese, remove the pan from the oven, spread the cheese evenly over the top, and place back in the oven for 3-4 minutes until the cheese has melted. 

Serve chicken with onions and cheese on its own, or with a side salad or another green vegetable.