Combine first 3 ingredients in a bowl; cover with plastic wrap and vent. Microwave at high for 1 1/2 minutes; set aside.
Combine onions, oil, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a 13 x 9-inch baking pan; stir well to coat.
Bake at 450 for 30 minutes on bottom rack of oven.
Trim fat from roast; place on a rack in a shallow roasting pan.
Combine cracked black pepper, 1/4 teaspoon salt, and crushed garlic; rub over surface of roast. Insert a meat thermometer into thickest part of roast. Stir onions; place roast on next-to-bottom rack in oven.
Bake onions and roast at 450 for 15 minutes. Reduce oven temperature to 35
Bake an additional hour, stirring onions after 30 minutes. Stir apricot mixture into onions; continue baking onion mixture and roast 10 minutes or until thermometer reaches 16
Remove roast and onions from oven. Cover roast loosely with foil; let stand 10 minutes. Stir in brown sugar and remaining ingredients to onion mixture.