Preparation Cut onion in half crosswise. Slice one-half into 1/4" rings; set aside. Finely chop remaining onion. Pulse chopped onion, garlic, cornmeal, 3/4 cup zucchini, 2 Tbsp. beans, and 2 Tbsp. corn in a food processor until finely chopped, 30–45 seconds. Heat 1 Tbsp. oil in a large skillet over medium. Add chopped onion mixture and cook, stirring occasionally, until mixture begins to soften and thicken, 3–4 minutes. Add remaining 3/4 cup zucchini and cook, stirring occasionally, until zucchini begins to soften and become translucent, 3–4 minutes more. Transfer mixture to a large bowl and gently mix in millet, 1/4 cup basil, 3/4 tsp. salt, 3/4 tsp. pepper, and the remaining beans and corn until well combined. Form mixture into 4 (1"-thick) patties. Cover with plastic and freeze 20 minutes. Meanwhile, combine mayonnaise, lime zest, lime juice, and remaining 1/4 cup basil in a small bowl; set aside. Prepare a grill for medium-high heat or heat a grill pan or cast-iron skillet over medium-high; oil grate or pan. Toss reserved onion rings with 1 Tbsp. oil and remaining 1/4 tsp. salt and 1/4 tsp. pepper. Cook onion rings until softened and lightly charred, about 5 minutes. Brush patties all over with oil. Cook, flipping once, until medium brown and toasted, about 7 minutes on each side; add 1 slice cheese to each during last 3 minutes of cooking. Transfer onions and burgers to a plate. Toast buns until golden brown, about 30 seconds. Place 1 piece lettuce on each bottom bun and top with a burger. Top with onions. Spread 1 Tbsp. basil mayonnaise on each top bun, add smashed avocado, and close sandwiches. Cooks' NoteBurgers can be made 3 months ahead; tightly cover and freeze. To serve, cook frozen patties 10 minutes on each side, adding cheese during last 3 minutes of cooking.