Combine the chopped apricots and the apricot nectar in a small saucepan. Bring to a boil over medium-high heat. Reduce the heat and simmer briskly until reduced and somewhat syrupy, about 10 minutes. Pour into a French press and let cool to room temperature, 5 to 10 minutes. Then add the bourbon, vermouth and bitters; stir well.
Add ice to 4 rocks glasses. Place the lid on the press and plunge. Divide the drink between the glasses and garnish each with a whole dried apricot.
Don't throw out the chopped apricots--they are delicious and can be used in vinaigrettes, on ice cream, etc.