Begin by gathering and prepping all of your ingredients according to the ingredient list above to have ready and organized for use.
Place a large, deep, non-stick pan over medium-high heat, and add in about 1 tablespoon of olive oil. Once the oil is hot, add in the smoked sausage medallions, and caramelize them for a few minutes until they become a deep brown color. Remove the sausage from the pan with a slotted spoon, and set aside.
To the same pan, add in the butter along with another 1 tablespoon olive oil, and allow them to melt together. Add in the sliced onions and caramelize those until a rich, golden-brown, about 8 to 10 minutes.
Next, add in the garlic, a couple of pinches of salt, the freshly cracked black pepper, the paprika and smoked paprika, and stir to combine with the onions. Sauté just until the garlic becomes aromatic.
Then, add in the sliced potatoes and fold them into the caramelized onions/garlic to coat them well. Add in the chicken stock and stir to combine, then push the sliced potatoes down into the stock/onion mixture as much as possible.
Cover the pan with a lid askew to allow some steam to escape, and simmer on medium-low heat for about 15 minutes, stirring gently once or twice. Uncover and allow the potatoes to continue to simmer for another 10 minutes until tender and the sauce thickens, stirring occasionally.
Finish the goulash by adding the caramelized smoked sausage back into the pan, as well as the parsley, and gently fold those in to incorporate. Add a drizzle of olive oil and additional salt and pepper if necessary.
Serve with warm, crusty bread, if desired.