Brine pork: Simmer all brine ingredients in a large stockpot with 3 cups water, stirring occasionally, until salt dissolves. Remove from heat and add 9 cups cold water. Let cool. Add pork and chill, covered, at least 1 day and up to
Begin rub while pork is brining: Preheat oven to 375°. Slice garlic heads in half crosswise and set on a sheet of foil. Drizzle with about 1 1/2 tbsp. oil and enclose in foil. Roast 40 minutes, or until cloves are buttery soft. Let cool.
Remove pork from refrigerator about 45 minutes before cooking, drain, dry thoroughly with paper towels, and let sit at room temperature.
Pull oregano and rosemary leaves from stems; chop finely, along with parsley leaves. Crush lavender in a mortar with a pestle (or put in a resealable plastic bag and pound with a meat mallet). Squish garlic cloves from their skins into a small bowl. Add 1/4 cup oil and smash garlic into a paste with a spoon. Stir in herbs and lavender; season lightly with salt and pepper. Pat garlic herb paste all over pork.
Ignite 45 to 50 briquets in a chimney starter on a fireproof surface (not on firegrate). Set a drip pan (roughly 6 by 8 in.) in center of firegrate and fill halfway with water. When coals are coated with ash (about 15 minutes), use tongs to arrange on either side of drip pan. Set cooking grate in place. Cover grill and, if your grill doesn't have a built-in thermometer, insert a smoker thermometer* through lid vent. Close grill vents as needed to bring temperature down to 300° (you may need to remove coals to get the temperature down; return them to the chimney starter).
Put meat, fat side up, on cooking grate over drip pan and cover grill. Grill pork, keeping the temperature between 250° and 300° and adding 5 or 6 coals to either side of drip pan every 30 minutes or so, until an instant-read thermometer inserted into pork registers 160°, about 4 1/2 hours.
Let pork rest 15 minutes before carving.
*Find a smoker thermometer online at Amazon.com ($17).
Note: Nutritional analysis is per serving.