Make the apricot syrup: Toss the chopped apricots, sugar, and water into a small saucepan. Bring it to a simmer over medium heat, stirring until the sugar’s dissolved.
Turn the heat down and let it cook for about 5 minutes, until the apricots soften. Take it off the heat and let it cool for 15 minutes.
Strain everything through a fine mesh strainer, pressing on the solids to get all the juice out. Let the syrup cool off completely.
For each cocktail: Fill a highball glass with ice. In a cocktail shaker, combine bourbon, apricot syrup, cooled tea, and lemon juice.
Add a handful of ice to the shaker. Shake it hard for about 15 seconds—really go for it—until it’s nice and cold.
Strain into your prepared glass. Drop in an apricot slice and a mint sprig for garnish.
Give it a gentle stir before serving so all the flavors mingle.