Make the simple syrup: Combine the sugar and 1 cup water in a small saucepan and bring to a boil. Cook, stirring, until the sugar dissolves. Remove from the heat and let cool completely. (It will make about 1 1/2 cups. Store extra syrup covered in the refrigerator up to 1 month.)
Make the tea: Steep all the tea bags in 3 cups hot water for 10 minutes. Remove the bags and add 2 cups cold water. Put the chopped peach, 3/4 cup simple syrup and the lemon juice into a blender and puree until smooth. Strain the mixture through a fine-mesh strainer. Stir together the peach-simple syrup mixture and the iced tea. Pour half of the peach tea into ice cube trays and freeze until solid. Refrigerate the remaining peach tea.
Make the cocktails: Fill 6 tall glasses with the peach tea ice cubes. Divide the peach tea evenly among the glasses. Top each with the prosecco and a lemon slice and serve.