Mix 10 tablespoons brown rum with 10 tablespoons brown sugar. Halve the peaches and remove the stones. Place the peach halves cut-side down in the rum mixture and marinate for 20 minutes.
Finely grate the toast slices into breadcrumbs using your fingers. Coarsely crumble the amarettini with your fingers. Heat a pan over medium heat. Add the breadcrumbs and almonds, roasting for 2-3 minutes until golden brown. Add the crumbled amarettini and roast for another 2 minutes. Remove from heat and let cool.
Remove the peach halves from the marinade and let drain. Place on a hot grill over medium-high heat, cut-side down, and grill for 5 minutes.
Arrange the grilled peach halves on plates. Spread the cooled amarettini mixture on top. Drizzle with the remaining rum mixture from the marinade.