Slice the mozzarella into 1/2-inch thick slices. You should have about 12 slices.
Place the flour in a small bowl and season with 1/4 teaspoon of the salt and a pinch of the pepper. Place the bread crumbs in another bowl and season with 1/2 teaspoon of the salt and a pinch of pepper. Combine the egg and milk in a third small bowl, season with the remaining 1/4 teaspoon salt and a pinch of pepper, and whisk to combine. Working 1 piece at a time, dredge the mozzarella slices first in the flour, then in the egg mixture, then finally in the bread crumbs, shaking between stages to remove excess coating. Place the breaded slices on a large plate and set aside; repeat with remaining cheese slices.
Add enough vegetable oil to a large skillet to come about 1/2-inch up the sides. Heat the oil over medium heat until hot. When hot, add the cheese slices and fry until golden on both sides, about 1 minute total. Using a slotted spoon or strainer, transfer briefly to paper towel lined plate to drain. Repeat with remaining cheese slices.
Combine the lettuces, tomatoes, and red onions in a mixing bowl and add dressing, to taste. Toss gently but thoroughly to coat well. Divide the salad evenly among 6 entree plates or large salad plates. Place 2 cheese slices against the side of the salad and serve immediately.
In blender, combine first 7 ingredients and blend until smooth. While the blender is still running, add the oil in a thin, steady stream until completely incorporated and emulsified.