Place the flour in a resealable plastic bag and season with the garlic, 1 teaspoon salt and 1/4 teaspoon pepper. Place the chicken pieces in the bag with the flour and toss to coat (work in batches). Transfer the breadcrumbs to a rimmed plate, season with salt and pepper and set aside.
Whisk the egg and 1 tablespoon water in a medium bowl. Remove a piece of chicken from the flour, dip in the egg mixture, then the breadcrumbs, gently pressing the crumbs into the chicken, and set on a clean plate. Repeat with the remaining pieces of chicken.
Heat the vegetable oil in a large frying pan over medium-high heat. Add the chicken pieces to the pan (work in batches if needed) and brown on each side, about 8 minutes total. Use a spatula to transfer the chicken nuggets to a paper-towel-lined plate. Salt while hot and serve.