Prepare and cook the apple sausage patties: Grate the Honeycrisp apple and gently squeeze out excess moisture. Mix with the hot Italian sausage, honey, and fennel seeds, then form into two patties slightly larger than the brioche buns. Heat a skillet over medium heat with a little olive oil and cook for 4–5 minutes per side, until browned and the internal temperature reaches 160°F.
Toast the brioche buns: Spread a thin layer of butter onto the cut sides of the brioche buns. Toast in a skillet over medium heat until lightly golden brown. Remove from the pan and set aside.
Cook the folded eggs with smoked Gouda: Whisk the eggs with salt and pepper. Melt half the butter in a nonstick skillet over medium-low heat and pour in half the egg mixture, tilting the pan to create an even layer. Once mostly set, sprinkle half the smoked Gouda down the center and fold the sides inward over the cheese. Cook until melted, then repeat with the remaining eggs, butter, and cheese.
Assemble: Spread a layer of Dijon and a layer of pepper jelly on each piece of brioche. Place a cooked apple sausage patty onto the bottom half of each brioche bun. Add the folded smoked Gouda eggs, then finish with the top bun. Serve immediately while the eggs are warm and the cheese is melted.