Heat oil in a frying pan over medium-high heat, add pumpernickel and fry, turning with a spatula, until crisp (6-8 minutes). Drain on paper towels and leave to cool.
Wipe pan clean, then cook jamón in batches, turning once, until crisp (2-4 minutes). Cool briefly, then break into shards.
For hazelnut vinaigrette, whisk lemon juice and oil in a bowl and season to taste, then stir in hazelnuts.
Place peaches in a bowl, add half the dressing and toss to combine, then layer on a serving platter with pumpernickel croûtons, jamón, mint and basil. Drizzle with extra dressing over or serve it to the side.