Make the brine: Bring 2 cups water, 3/4 cup salt and the peppercorns to a simmer in a large saucepan over medium-high heat, stirring to dissolve the salt. Let cool, then stir in the apple and lemon juices.
Cut the backbone out of each chicken using kitchen shears; discard. Trim off any remaining ribs. Place the chickens breast-side up, then firmly press down with your hands to flatten. Trim off any excess skin. Transfer the chickens to a large pot, laying them flat (or use a two-gallon resealable plastic bag). Add the brine, thyme, garlic and 2 cups cold water. (If the chickens are not covered, add 1 to 2 more cups water.) Refrigerate 8 hours or overnight.
Place a drip pan under the grill grates. Preheat the grill to medium and prepare for indirect cooking: On a gas grill, turn off half the burners; on a charcoal grill, bank the coals to one side. Meanwhile, soak the applewood chips in water, 30 minutes; drain. Remove the chickens from the brine and pat dry. Season on both sides with the paprika, 1 teaspoon salt and 1/2 teaspoon pepper.
When the grill registers 375 degrees F, add the wood chips: On a gas grill, fill a smoke box with half the chips and use according to the manufacturers' instructions; on a charcoal grill, sprinkle half the chips over the coals. Place the chickens breast-side up on the cooler side of the grill, making sure they lie flat. Cover the grill and cook 1 hour, then add the remaining wood chips (if using charcoal, add more coals at this point or as needed so the temperature stays between 325 degrees F and 375 degrees F). Continue grilling until a thermometer inserted into the thighs and breasts registers 165 degrees F and the skin is browned and crisp, 1 to 1 1/2 more hours. Transfer the chickens to a cutting board and let rest at least 15 minutes before carving.