Whisk together buttermilk, eggs, garlic cloves, 2 teaspoons of the salt, and 1 teaspoon of the pepper in a large bowl. Place chicken in a large ziplock plastic freezer bag; pour buttermilk mixture over chicken. Seal bag, removing as much air as possible. (Or combine in a large bowl, and cover with a tight-fitting lid.) Chill 8 to 24 hours.
Preheat oven to 425°F. Pour oil into a large cast-iron skillet, and heat over medium-high.
Drain chicken well; discard buttermilk mixture. Whisk together flour, cornstarch, 2 teaspoons of the salt, and 1 teaspoon of the pepper in a large bowl. Working with 1 piece at a time, dredge chicken in flour mixture until well coated. Place coated chicken pieces on a large piece of aluminum foil, and let stand to allow coating to dry out, about 15 minutes.
Fry chicken, in batches, 4 pieces at a time, in hot oil until golden, about 3 minutes per side. Transfer browned chicken to a wire rack in a large rimmed baking sheet. Bake in preheated oven until chicken is deep golden brown and cooked through, about 10 minutes.
Meanwhile, let skillet rest off heat about 5 minutes while chicken bakes. Carefully pour all of the frying oil into a metal bowl; allow flour to settle to bottom of bowl. Wipe skillet clean.
Add 2 tablespoons of the frying oil to skillet, and place over medium-high. Add green beans and shallot, and cook, without stirring, until beginning to brown, about 4 minutes. Stir in tomatoes, and cook, without stirring, until beans and shallot are charred and tomatoes begin to burst, about 2 minutes. Stir in remaining 3/4 teaspoon salt and 1/2 teaspoon pepper. Transfer vegetables to a serving platter, and toss with lemon juice. Serve the vegetables immediately with the fried chicken.