Place onions on a microwave-safe plate and cook in the microwave to soften about 1 minute.
Place chicken breasts on a plate and squeeze the juice of 1 lemon all over the chicken. Let sit for about 5 minutes. Combine thyme, garlic powder and ¼ teaspoon salt and sprinkle ½ of these seasonings over top of chicken.
Heat oil in a large nonstick pan, coated with cooking spray. Add softened onions. Cook for 1 minute over medium-high heat. Add chicken and the lemon juice to pan of onions, seasoning side down. Sprinkle the top of chicken with remaining seasonings. Sauté over medium-high heat, about 5 minutes, turn over, cover, and continue to cook over medium heat, about 4 minutes, until done. Remove chicken from pan to a serving plate.
Turn heat down to medium-low. Add apricot preserves, dried apricot pieces, balsamic vinegar, ¼ teaspoon salt, and a little pepper. Mix well to combine. Stir constantly until the preserves melt, about 30 seconds or so. Spoon apricot sauce over chicken.