Trim visible fat and tendons from the chicken breasts and make small slits down the length of each breast to help the marinade penetrate.
In a bowl, whisk together the lemon juice, garlic puree, dried poultry seasoning, and 4 tablespoons olive oil until combined.
Place the chicken in a large resealable plastic bag or a lidded container in a single layer, pour in the marinade, seal, and turn to coat all pieces.
Refrigerate the chicken to marinate for 6–8 hours, or up to 24 hours if preparing ahead.
While the chicken marinates, prepare the Roasted Red Pepper Aioli Sauce and chill it in the refrigerator.
When ready to cook, remove the chicken from the fridge and let it come to room temperature while you preheat the grill to medium-high.
Oil the grill grates or brush the grill pan with olive oil or use nonstick spray to prevent sticking.
Grill the chicken breasts 4–5 minutes on the first side; optionally rotate once for grill marks.
Flip the chicken and grill 4–5 minutes more on the second side, or until the chicken is firm, lightly browned, and cooked through.
Serve the grilled chicken hot with the chilled Roasted Red Pepper Aioli Sauce spooned over or served alongside.