Over medium heat, cook shallots and ginger in sesame oil in a small heavy saucepan over medium heat, stirring occasionally, until softened and golden, about 3 minutes. Add garlic, stir, and continue to cook another 2 minutes.
Stir in vinegar and sugar and boil until reduced by about half, about 2 minutes.
Add soy sauce and apricots and simmer, uncovered, stirring occasionally, 15 minutes.
Purée sauce in a blender until smooth (use caution when blending hot liquids), then cool to room temperature.
Place chicken in a large sealable bag and pour in marinade. Seal bag, pressing out excess air, and marinate, chilled, turning bag over occasionally, at least 8 hours (I did a full 24 hours).
Preheat oven to 400°F with rack in middle.
Arrange chicken in 1 layer in pan. Roast chicken for 15 minutes. Turn, spoon any excess sauce on top of chicken, and cook another 10 minutes. Set broil to high, and broil until tops are golden brown and the sauce has seized, 4-5 minutes.