Make the barbecue sauce: in a heavy nonreactive saucepan, add the ketchup, vinegar, mustard, Worcestershire sauce, lemon, onion, brown sugar, chili powder, garlic salt, black pepper, ginger, cayenne, and 1 cup water; slowly bring to a boil over medium heat.
Decrease heat to medium and let the sauce simmer gently until thick and full-flavored, about 30 minutes, adding a little more water if it gets too thick.
Taste for seasoning, adding more brown sugar, garlic salt, and/or even vinegar, as necessary, to taste.
Strain the sauce before serving, if desired; transfer the sauce to clean jars and refrigerate until serving.
Make the rub: combine the rub ingredients in a bowl; use your fingers to mix the rub; set aside roughly 1/3 of the rub.
Trim the outer layer of fat on the brisket so that about ¼ inch remains; if there is a pocket of fat between the flat and the cap, cut it out and discard it.
Cut the meat into 3 equal-size pieces; sprinkle 2/3 of the rub all over both sides of the pieces of brisket, dividing it evenly among them and patting it into the meat with your fingers.
Wrap meat in plastic wrap and let cure 12 hours in the refrigerator.
Pour the cider into a spray bottle; set aside.
Set up grill for indirect grilling and preheat to low; **if using a gas grill, place all the wood chips in the smoker box or in a smoker pouch and run the grill on high until you see smoke, then decrease heat to low; **if using a charcoal grill, place a large dip pan in the center, preheat grill to low, then toss 1 cup of wood chips on coals; **if using a smoker, fire it up according to manufacturers directions.
When ready to cook, place the brisket in the center of the hot grate, fat side up, over the drip pan and away from heat; cover the grill.
Cook brisket until handsomely browned, about 3 hours.
Spray the meat every hour with apple cider.
Turn over the meat and cook 1 more hour.
**If using a charcoal grill, every hour you will need to add 12 fresh coals and ½ cup of wood chips to each side.
Spray each piece of meat one more time with apple cider; sprinkle the reserved rub all over the pieces of meat, dividing it evenly among them, then wrap each piece in heavy foil.
Return the wrapped meat to the grate, cover the grill, and continue cooking the brisket until very tender, 3-4 hours; **if using a charcoal grill, you will need to add 12 fresh coals to each side every hour, no more chips are necessary.
Partially unwrap one of the briskets to test for doneness—should be very dark and tender and have an internal temperature of 190°.
Transfer brisket to a cutting board; let rest for 20 minutes.
You can serve brisket right away—slice it thinly across the grain; or you can refrigerate it overnight and reheat it in a 350° oven for about 1 hour.