For the onions: Heat a large oven proof 10-inch nonstick pan, with oil in it, to medium heat. Add the onions and season with salt and pepper. Cook until the onions are soft and lightly caramelized, about 10 minutes. Once they are caramelized, pour them into a bowl and set aside.
For the giardiniera frittata: Set the oven to broil. Whisk the eggs in a bowl. Wipe the skillet you cooked the onions in and place back over medium-low heat, then add the butter and olive oil. Add the eggs and slowly cook until the bottom is set but the top is still wet, 4 to 6 minutes. Sprinkle the Hot Giardiniera evenly over the eggs and season with salt and pepper.
Place the skillet in the broiler on the middle rack and broil until the top is spotty brown and puffy, about 5 minutes. Let the frittata cool slightly and, using a 3- to 4-inch ring mold, cut out circles that will match the size of your bread.
For the burger: Heat a grill pan to medium-high heat. Grill each side of bologna until grill marks form, 3 to 4 minutes a side. Once you flip to the second side, top each slice with a piece of cheese and cover with a metal bowl until melted.
For the sandwich build: Layer the bottom bun with a dollop of the Dusseldorf mustard, bologna with cheese, onions and the frittata. Schmear the top bun with another dollop of Dusseldorf mustard. Close the sandwich, recite the Pledge of Allegiance and eat whole.
Homemade Hot Giardiniera:
Combine 2 cups water and the salt in a glass or non-reactive bowl. Mix until the salt is dissolved. Add the carrots, cauliflower, serranos, garlic, celery and bell pepper to the salt water and stir to combine. Cover and refrigerate overnight.
Day 2, drain and rinse the vegetables. In a clean bowl, mix together the oil with the oregano and pepper. Add the vegetables and mix to combine. Allow to marinate overnight. Giardiniera will only get better with time. After 2 days at the most in the bowl, you can place in air-tight mason jars and keep in the fridge for at least 2 to 3 weeks.