In a food processor or blender combine lemon thyme and pistachio nuts; cover and process or blend with several on/off turns until finely chopped.
With procesor or blender running, gradually drizzle in oil, stopping to scrape down sides as necessary.
Stir in freshly ground black pepper to taste.
For a charcoal grill, arrange medium hot coals around a drip pan.
Test for medium heat above pan.
Place chicken, bone side down, on a grill rack over drip pan.
For gas grill place chicken adjacent to heat source.
Cover and grill for 50 to 60 minutes or until chicken is no longer pink (instant read thermometer reads 180°C for breasts or 180°C for thighs and drumsticks), turning chicken and brushing with half the pesto during the last 5 minutes of grillling.