In a mini food processor, blend together lemon juice, oregano, garlic, salt, pepper, then slowly add oil while the processor is running to make the sauce.
To avoid contamination divide the sauce into two separate bowls. One bowl with 4-6 tablespoons for the marinade, and the other bowl with the remaining sauce.
Place the breasts in a baking dish or resealable bag and add 1 tablespoon of marinade sauce for each breast. Brush or rub the sauce to coat each piece. Place chicken in the fridge (covered) for 30-60 minutes.
Starting with a clean grill; preheat all burners on medium (medium-high) heat. Adjust heat to reach approximately 400°F.
Arrange chicken breasts, bone side down (skin up), on top rack of grill.
Cook, with the lid closed, changing positions of the breasts twice, until cooked through, about 40 minutes (this will vary based on size and thickness of breasts, as well as actual temperature of grill). Remove chicken from the grill, when the internal temperature reaches 160-162°F.
Transfer chicken to a plate, pour a little sauce over each breast, loosely cover with foil, rest for 5-10 minutes. Internal temperature with rise to 165°F while resting.
Grill lemon slices, until grill marks appear, about 2-3 minutes/side. Serve each breast with 1-2 lemon slices and remaining sauce.
Cook’s note: I find cooking chicken on the upper rack (of a clean grill), takes the stress of flares up away. The occasional flame will add a little char and flavour. Also, the heat from the top of the grill will brown and crisp up the skin. However, if the skin doesn’t brown in your grill, put the breasts skin side down on the lower rack for 3-4 minutes, until desired colour is attained.
Also delicious using chicken legs! Cook legs to 165°-167°F, before removing from grill.