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In a small saucepan, melt the butter over medium heat. Add the preserves, vinegar, red pepper flakes, rosemary, and a large pinch of salt; stir to combine. Bring to a medium boil, reduce the heat to low and simmer, stirring occasionally, until slightly thickened, 2 to 4 minutes. Remove from the heat and let cool to room temperature. (If making ahead, store covered in the refrigerator. Before using, warm over low heat to loosen the consistency.)
Prepare a medium gas or charcoal grill fire. Using a stiff wire brush, scrub the cooking grate thoroughly. Dip a folded paper towel into vegetable oil and, using tongs, rub it over the grill grate.
Season the chicken lightly with salt and pepper. Set the parts skin side down on the grill. Cook, covered, until the skin is golden brown, about 10 minutes. Stay near the grill, especially during the first 10 minutes, to manage any flare-ups, by moving pieces out of the way. If the chicken is browning too quickly, turn the heat down slightly or close the vents partially. Flip the chicken and cook until an instant-read thermometer reads 165°F in the thickest part of each piece, 5 to 10 minutes more. The thighs, legs, and thinner breast pieces are apt to cook a little faster than the thicker breast pieces. Transfer each piece to a platter when done and tent with foil.
When all the chicken is done, brush it with the glaze on all sides. Return the chicken to the grill and cook for another minute or so on each side to caramelize the glaze. Brush the chicken with any remaining glaze and serve.
Make Ahead Tips
The apricot glaze can be made up to a day ahead and stored covered in the refrigerator. Before using, warm over low heat to loosen the consistency.