Whisk the mayonnaise, pepper brine, mustard, rosemary and 1 tablespoon water in a small bowl. Season with salt and pepper. Spread evenly on the cut sides of the rolls. Heat 1 1/2 inches olive oil in a pot over medium-high heat until a deep-fry thermometer registers 350 degrees F. Line a baking sheet with paper towels. Whisk the flour, 1 teaspoon salt and 1/2 teaspoon pepper in a large bowl. Whisk the eggs and 2 tablespoons water in another bowl. Toss the onions in the flour, then fry in the hot oil until crisp, about 1 minute. Transfer to the prepared baking sheet to drain. Toss the mushrooms in the flour, dip in the egg mixture and toss again in the flour; fry in batches until golden, about 2 minutes. Drain on the baking sheet and season with salt and pepper. Fill the rolls with the ham, Peppadew peppers, and fried onions and mushrooms.