Remove mint leaves from stems. Place mint leaves in bowl and cover with 1/2 cup bourbon. Soak for 15 minutes. Remove leaves from bowl of whiskey. Wrap leaves in paper towel and wring over bowl. Set mint leaves aside.
To make simple syrup, mix 1 cup granulated sugar and 1 cup water in small saucepan. Heat over medium heat until sugar dissolves, stirring constantly. Set aside to cool.
Pour 3-1/2 cups bourbon into a large pitcher. Add 1 cup of simple syrup to bourbon in pitcher. Add small bowl of mint-infused whiskey to pitcher. Muddle remaining mint leaves with jalapeno slivers. Mix jalapeno and mint into pitcher and refrigerate for at least an hour. Add ice to pitcher and serve cold.