In a medium bowl, whisk together the eggs and crème fraîche until smooth.
Season with salt and black pepper.
Slice the bagels in half.
Spread mayonnaise on one side of each bagel and Dijon mustard on the other side.
Heat a non-stick skillet over medium-high heat.
Add the butter and let it melt until lightly foaming.
Pour the egg mixture into the skillet, tilting the pan to spread it into a thin layer.
Cook until the edges begin to set while the center remains slightly soft.
Place the bagel halves face-down onto the eggs.
Carefully flip the egg sheet so the bagels sit underneath.
Fold any excess egg over the bagels using a spatula.
Remove from the skillet and set aside.
Add a drizzle of vegetable oil to the same pan.
Fold the mortadella slices loosely and place them in the skillet.
Cook for 2–3 minutes per side until golden and crisp around the edges.
Sprinkle mozzarella cheese over the mortadella after flipping.
Allow the cheese to melt completely.
Return the egg-covered bagels to the skillet for about one minute to warm through.
Layer the cheesy mortadella onto the egg-covered bagel bottoms.
Top with the remaining bagel halves.
Serve immediately while warm and melty.