How to Make It
Whisk together 8 cups warm water, 2/3 cup salt, brown sugar, and 1 tablespoon lemon juice in a large bowl or pot until salt and sugar dissolve. Add turkey and chicken. Cover and chill 1 hour and 30 minutes. Place a colander in sink; drain meat in colander, rinse with fresh water, and set meat aside.
While meat chills, process onion, garlic, and jalapeño in a food processor until very finely chopped, about 10 seconds. Heat oil in a medium skillet over medium. Add onion mixture and a pinch of salt; cook, stirring often, until softened, 5 to 7 minutes. Stir in parsley, oregano, and rosemary; cook, stirring constantly, 1 minute. Remove from heat; transfer sofrito to a small bowl. Chill, uncovered, 20 minutes.
Stir together mayonnaise, 1 teaspoon salt, lemon zest, and remaining 2 tablespoons lemon juice until smooth. Cover and refrigerate until ready to use.
Preheat grill to high (450°F to 500°F). Working in 4 batches, place brined meat in a food processor; pulse until mixture forms a coarse paste, about 7 (2-second) pulses. Transfer to a medium bowl. Add sofrito; mix with hands until thoroughly combined. Shape into 6 tightly packed 7-ounce patties. Grill, uncovered, flipping often, until a thermometer registers 155°F, 10 to 13 minutes. Top each patty with about 1/4 cup cheese. Grill, covered, until cheese melts, about 2 minutes.
Season tomatoes with salt and pepper. Spread each bun with 1 teaspoon aioli and 1 teaspoon barbecue sauce. Top each bottom bun half with 1 patty, 1/4 cup lettuce, and 1 tomato slice. Cover burgers with top bun halves.
Uncooked burger patties may be individually wrapped and chilled 1 day or frozen up to 1 month. Thaw in the refrigerator before proceeding.