Add the mint leaves and peach-jalapeño syrup to a shaker, and crush the mint and syrup with a muddler to release the oils.
Fill the shaker almost to the top with ice.
Add the cognac and lemon juice, then shake and strain into a small ice-filled Mason jar.
Garnish with diced peaches and jalapeño slices.
*Peach-jalapeño syrup: Combine 1 cup sugar, 1 cup water and 1 sliced jalapeño in a saucepan, and heat until the sugar dissolves. Let sit for 1 hour or up to overnight, then strain. Repeat the same process with 1 cup sugar, 1 cup water and 1 diced peach. Combine the syrups once cooled, and store in refrigerator until ready to use.