Heat 2 cups neutral oil in a large high-sided or cast-iron skillet (at least 10 inches) over medium-high heat until 350ºF. Meanwhile, assemble the sandwiches and prepare the egg mixture.
Place 4 slices white sandwich or milk bread on a cutting board. Spread 1 tablespoon peanut butter on each slice, leaving a ½-inch border. Top each with a second slice of bread to close the sandwiches. Cut the crusts off each sandwich. Place 2 large eggs, 2 tablespoons whole milk, and 1 pinch kosher salt in a wide, shallow bowl (such as a pie dish), and whisk until combined.
When the oil is hot, fry 2 at a time: Dip all the sides of each sandwich quickly in the egg mixture (do not let it soak) and place in the hot oil. Fry, flipping halfway through, until deep golden brown all over, about 1 ½ minutes per side. Transfer to a paper towel-lined plate to drain briefly, then place each sandwich on a plate. Repeat dipping and frying the remaining 2 sandwiches.
Top each French toast with 1 tablespoon sweetened condensed milk and ½ tablespoon unsalted butter. Serve immediately.
