Pat mozzarella thoroughly dry and let it sit on fresh paper towels for 10 minutes. Put flour in one shallow bowl, beat egg with water in another and mix panko with salt in a third.
Coat each cheese slice in flour, then egg, then panko, pressing crumbs gently onto all surfaces. Set on a plate and freeze for 20 minutes to firm the coating without freezing the cheese solid.
Heat oven to 300°F and warm croissant halves cut-side up for 5 minutes. Lay tomato slices on paper towels and blot away surface moisture.
Heat oil in a 10-inch skillet over medium heat to 350°F. Fry mozzarella in two batches for 1 to 2 minutes per side until the coating is golden and crisp; transfer to a rack and rest for 2 minutes.
Mix olive oil, balsamic vinegar and pepper. Brush lightly over the cut faces of the warm croissants.
Place one fried mozzarella slice on each bottom, add two tomato slices and divide basil leaves over them. Close gently and serve immediately while the cheese coating is crisp.