Directions

1. Whisk 1 1/2 cups pesto, 1/2 cup oil, 1 tsp. salt, and 1/2 tsp. pepper in a medium bowl until smooth. Add chicken to bowl and slide your hand between skin and meat to loosen skin. Rub half of pesto mixture under skin, then half over outside. Cover bowl with plastic wrap and chill at least 1 hour or up to overnight.
2. Prepare a grill for medium-high heat or heat a grill pan or cast-iron skillet over medium-high; oil grate or pan. Season chicken with remaining 3/4 tsp. salt and 1/4 tsp. pepper. Grill chicken, skin side down, covered, until golden brown and lightly charred, 7–9 minutes. Turn and grill until cooked through and an instant-read thermometer inserted into thickest part of thigh registers 165°F, 5–7 minutes. Transfer chicken to a plate.
3. Brush peach halves with oil. Reduce heat to medium and grill peaches, cut side down, until warmed through, 2–3 minutes.
4. Meanwhile, whisk remaining 1/4 cup pesto and 2 Tbsp. oil in a small bowl. 
5. Arrange chicken and peaches on a platter and drizzle pesto sauce over. Serve remaining sauce alongside.