Lay the milk bread slices side by side on a cutting board. Spread 1 tablespoon of peanut butter on each of 2 slices and spread 1 tablespoon of sweetened condensed milk on each of the remaining 2 slices. Sandwich each peanut butter slice with a sweetened condensed milk slice, filling-side in. Trim the crusts to make an even square.

Fill a medium saucepan with 1 inch of oil and heat over medium heat until it registers 350 degrees F on a deep-frying thermometer. Prepare a large paper towel-lined plate.

Whisk the eggs in a shallow dish until well beaten. Submerge one sandwich in the egg, turning until all sides are fully coated. Set aside on a clean plate and repeat with the remaining sandwich.

When the oil is ready, carefully transfer one sandwich to the pan and fry until golden brown on the bottom, about 2 minutes. Flip and fry until the other side is golden brown, about 90 seconds more.

Transfer the sandwich to the prepared plate with a slotted spoon and allow it to cool slightly and the excess oil to drain off, about 30 seconds.

Transfer to a serving plate and top with 1 tablespoon of butter. Drizzle with additional condensed milk. Enjoy while warm. Repeat with the remaining sandwich and butter.