Place butter and blueberries in a food processor and pulse 4-5 times to blend, then process for 30 seconds.
Transfer to a small bowl and fold in the chopped basil.
Cover with plastic wrap and refrigerate until ready to use.
Brush the cut sides of the peaches with oil and place on the grill, cut side down.
Grill for 2-3 minutes on each side or until they begin to soften. Place on a platter.
Top each peach with 1/2 tablespoon of the blueberry basil butter and serve immediately.