First, peel and thinly slice the onion into half-moon shapes. If you slice it length-wise before slicing it, you will get half-moon shaped slices.
In a cast iron pan (or other oven-safe skillet), melt 1 tablespoon of the butter and 1 tablespoon of the olive oil over medium-high heat.
Add the sliced onions to the pan and cook while stirring occasionally until golden brown. This will take about 20-25 minutes; lower the heat if necessary if the onions threaten to burn.
When the onions are done, remove them from the skillet and turn off the heat. Don’t wash the skillet.
Meanwhile, crumble the Cashel blue cheese and have it ready in a small bowl.
Toast the walnuts if they’re not already toasted, chop, and have them ready in a small bowl.
Preheat the oven to 400.
Take each chicken breast, and turn it on its side. Place the thickest side down. Using a very sharp knife, cut a slit into the breast. Be careful not to cut all the way through, yet try to make your slit as large as possible to hold as much filling as possible.
Sprinkle each chicken breast very well with salt and pepper. Then, divide the blue cheese between each breast, and stuff it as far back as possible (this prevents it from melting and oozing out in the oven).
Next, push as many caramelized onions as will fit into each breast; you may have some leftover caramelized onions, depending on the size of your onion.
Finally, press a few walnuts into the onion and Cashel mixture.
Next, put the pan you cooked the onions in back on the stove, and turn it to medium-high heat. Place the chicken breasts in the pan, and cook on each side until golden brown. Don’t worry about cooking them all the way through, they will finish in the oven.
Once the chicken breasts as golden brown on both sides, move the pan to oven. Cook for 15 minutes, and then test the thickest part of the breast with a meat thermometer to ensure it reaches 165-degrees F.
Serve with mashed potatoes.