Mix olive oil, vinegar, herbs, garlic, shallots, salt and pepper in a large bowl. Coat turkey pieces well.
Pieces can be refrigerated for up to 24 hours before grilling.
Turn one section of the barbecue off, preferably the middle one if you have 3 sections.
Place turkey thighs and drumsticks directly on this section of the grill to cook over indirect heat, leaving the other sections on high.
Wrap the boneless skinless breast loosely in parchment paper or foil and place it directly on the same part of the grill. Cook with the lid down on the barbecue. Drumsticks and thighs should be turned once after 30 minutes.
Pieces will be done in 45-60 minutes or when a digital thermometer probe reads 170F (77C).
Let cooked turkey pieces rest (cover with a loose tent of foil) in a warm place for 15-20 minutes before carving.
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