A quick brine produces super-moist burgers with chin-dripping juices. The brining also helps the burgers cook faster, so flip them often to help them cook evenly and prevent charring.
8 cups warm water
2/3 cup plus 1 teaspoon kosher salt, divided, plus more
1 tablespoon light brown sugar
1 tablespoon lemon zest plus 3 tablespoons fresh lemon juice (from 2 lemons), divided
1 1/2 pounds boneless, skinless turkey pieces (such as breasts, thighs, and drumsticks), cut into 1-inch pieces
1/2 pound boneless, skinless chicken thighs, cut into 1-inch pieces
1 1/4 cups chopped yellow onion
1/4 cup chopped fresh garlic
2 tablespoons chopped jalapeño
3 tablespoons extra-virgin olive oil
1 tablespoon finely chopped fresh flat-leaf parsley
1 tablespoon finely chopped fresh oregano
2 teaspoons finely chopped fresh rosemary
1 cup mayonnaise
6 ounces English cheddar cheese, grated (about 1 1/2 cups)
6 (1/4-inch-thick) tomato slices
Freshly ground black pepper
6 hamburger or brioche buns, split
2 tablespoons bottled barbecue sauce
1 1/2 cups shredded iceberg lettuce
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.
