Lay out two slices of brioche bread.
Place a slice of Swiss cheese on each.
Add two slices of ham on top of the cheese.
Place another slice of Swiss cheese on top of the ham.
Top with the remaining bread slices, pressing gently to hold everything together.
In a shallow bowl, whisk together the eggs, milk, salt, and black pepper until smooth.
Heat a skillet over medium heat and add the butter and vegetable oil.
Dip one sandwich into the egg mixture, coating both sides evenly.
Let any excess batter drip off before placing it in the skillet.
Repeat with the second sandwich.
Cook for 3-4 minutes on one side until golden brown and crispy.
Carefully flip and cook for another 3-4 minutes until the cheese is melted and the sandwich is golden.
Remove from the skillet and let it rest for a minute before cutting.
If desired, dust with powdered sugar and serve with raspberry jam.
Enjoy warm for the best texture and flavor.