Using a food processor, add in the 4 peaches (pits removed), sugar and lime juice. Pulse until pureed. Pour the contents into a fine sieve over a bowl to catch the nectar. Using the back of a spoon, push through all the nectar you can and toss the bits that don’t make it through.
[To make Rosemary Simple Syrup] Combine 2 cups of water, 2 cups of sugar, 3 tbsp. corn syrup and 1/4 cup rosemary leaves in a medium saucepan. Bring to a simmer, stirring until sugar dissolves. Remove from heat and rest for 20 minutes. Strain into a glass and refrigerate until chilled.
Using a cocktail shaker filled with ice, add in vodka, lime juice, honey, 2 tbsp. rosemary simple syrup, 4 tbsp. peach nectar and shake. Pour into two cocktail glasses filled with ice and top with Sprite and a splash of water. Garnish with two peach slices per drink and a rosemary sprig.