Gently press the tilapia pieces into the flour to coat; shake off the excess flour. Dip each flour-coated piece into the beaten egg mixture and then press into bread crumbs. Gently toss between your hands so any bread crumbs that haven't stuck can fall away. Place the breaded tilapia onto a plate while breading the rest; do not stack.
Heat the vegetable oil in a frypan over medium-high heat. Cook the breaded tilapia in the vegetable oil until golden brown, about 5 minutes per side.