Make brine: In a large stockpot, bring 4 qts. water to a boil with salt, brown sugar, garlic, lemongrass, ginger, and pepper, stirring often. Set pot in a sink of ice water and cool to room temperature. Stir in calamansi juice, then add chicken (or transfer to a large bowl if pot is too small). Set a plate on chicken to keep it submerged. Chill, covered, 1 to 2 days.
Meanwhile, make basting sauce: Heat oil with annatto in a small saucepan over low heat, stirring often, until oil is deep orange and flavorful, 3 to 5 minutes. Remove from heat and stir in lemon zest, salt, and pepper. Let cool, then chill, covered.
Make toyomansi sauce: Combine all ingredients in a squeeze bottle or small bowl; chill, covered, until used.
Prepare a grill for indirect medium heat (350° to 450°). For charcoal: Ignite a full chimney of briquets (about 90) on firegrate. When coals are dotted with ash, in 20 minutes, bank evenly on opposite sides of firegrate and if needed, let burn to medium. Set a 9- by 13-in. foil drip pan in center and set cooking grate in place. For gas: Remove cooking grates. Turn all burners to high, close lid, and heat 10 minutes. Turn off center burner(s) and reduce heat for others to medium. Set a 9- by 13-in. foil drip pan on turned-off burner. Set cooking grates in place (if drip pan sticks up, push on grates to flatten pan a bit).
Pour basting sauce through a fine strainer. Lift chicken from brine (discard brine) and set over indirect heat. Brush all over with about one-third of basting sauce. Grill, covered, 20 minutes. Baste chicken, then turn over and baste other side. Grill, covered, basting one more time, until no longer pink in thickest part, 20 to 30 minutes more (10 minutes for any separate wing pieces); during last 10 minutes, move chicken to direct heat to brown more.
Squeeze or brush toyomansi sauce all over chicken. Transfer chicken to a cutting board. Cut off legs and wings and quarter each whole breast through bones.
*Unless you have a friend with a calamansi (aka calamondin) tree, you're unlikely to find this small, tart citrus that tastes like a cross between lime and kumquat with a faint floral bitterness. (Though Seafood City supermarket chain in California sometimes carries fresh calamansi.) So go online for bottled Sun Tropics pure calamansi (amazon.com). Or use our Test Kitchen fresh citrus substitute: Combine 3 tbsp. lime juice, 1 tbsp. Meyer lemon juice, and 1 tbsp. orange juice.
Ask a butcher to prep whole chickens, or do it yourself. Annatto--red seeds of the achiote tree--add color and a deep earthy flavor; find at Latino markets and at worldspice.com.