In a medium bowl, whisk eggs and milk and set aside. Pour panko crumbs into a separate medium bowl and set aside. Dip both sides of bread slice into egg batter. Roll into breadcrumbs a few times until both sides are evenly and fully coated in breadcrumbs. Repeat with remaining bread.
Add butter to skillet and turn on the heat. After butter is melted, place bread onto skillet and cook until panko crumbs turn a golden brown. Flip over and repeat with the other side. Remove from pan and set aside. Repeat with remaining bread.
Place 2 slices of finished bread back onto skillet. Place a slice of cheese on each one and then place remaining 2 bread slice on top of the cheeses, forming 2 sandwiches. Pressing down slightly onto sandwich, heat both sides until cheese in the center is melted. Remove from heat and serve immediately.
The nutrition information provided are only estimates based on an online nutritional calculator. This is not a comprehensive list of all the nutrients in the recipe (i.e., does not include vitamins, cholesterol, etc). I am not a certified nutritionist. Please consult a nutritionist or doctor for accurate information and any dietary restrictions and concerns you may have.