In a medium saucepan over medium heat, add one tablespoon of olive oil. Once hot, toss in the thinly sliced sweet onions. Sauté for about 3 to 5 minutes until they are soft and translucent. This base is essential for creating depth in your sauce.
Once your onions have reached that beautiful translucent state, add the jalapeño slices and black peppercorns. Sauté for another 2 minutes. This step brings out the flavors of the jalapeños and gives your sauce that extra kick.
Next, pour in the cream. Switch to a low flame and stir gently. Allow the mixture to heat slowly for about 4 to 5 minutes. You want to see tiny bubbles, not a rolling boil.
Gradually whisk in the cheddar and American cheeses. Stir continuously until the cheese melts and blends smoothly into the cream. This might take about 5 to 7 minutes. The enchanting aroma coming from the pot will tempt you to dive in early.
Once the cheese is melted and the sauce is smooth, add sea salt, white balsamic vinegar, dry white wine, smoked paprika, and Dijon mustard. Stir these ingredients in thoroughly. Let it simmer for another couple of minutes, allowing the flavors to meld beautifully.
Pour the sauce into a serving bowl or use it straight from the pot. Drizzle it over your favorite burger, dip your fries, or even ladle it over fresh veggies. Enjoy the rich satisfaction of your homemade Shake Shack Cheese Sauce.