Cut the ends off peaches, then cut peaches horizontally into 1/4-inch-thick rounds, cutting inward until you reach the pit (like a donut shape). Turn peach to the opposite end and repeat.
Dice end pieces to yield 1 cup fresh peaches. Add diced peaches, olive oil, brown sugar, whiskey, sriracha, and lime juice to a small saucepan. Stirring constantly, bring just to a gentle boil. Remove from heat, stir in basil and lime zest, and allow to cool for 3 to 5 minutes.
Spray a grill pan with cooking spray; set over medium-high heat. Brush peach rounds with peach mixture.
Place peach rounds in the grill pan just long enough to caramelize a little and get good grill marks, 2 to 3 minutes per side.
On a platter, lay peach slices alternately with mozzarella slices, like shingles. Drizzle with remaining peach glaze and serve.
