Remove the peppers from the brine, reserving the brine.
Cut away the stems and top 1/4-inch from each pepper. Using a melon baler, scoop out the seeds.
Wrap a basil leaf around each bocconcini, and then wrap a strip of prosciutto around the basil. Place 1 wrapped ball inside each pepper. Season lightly with salt and pepper, and drizzle with oil.
Return the stuffed peppers to the brine. When ready to serve arrange on a platter and garnish with basil sprigs.
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