Preheat oven to 250 degrees. Place flour in a bowl. In another bowl, beat eggs. In a third bowl, combine panko with thyme. Season each with salt and pepper. Working with 2 bocconcini at a time, dredge in flour, shaking off excess, coat in egg, and roll in breadcrumbs, pressing gently to adhere.
In a small saucepan, heat vegetable oil over medium-high until pinch of breadcrumbs sizzles and floats to the top when added. Fry bocconcini, 4 at a time, until golden brown, turning occasionally, 2 minutes per batch. With a slotted spoon, transfer balls to paper towels to drain and season with salt. Keep warm in oven and repeat with remaining bocconcini. Serve with tomato sauce for dipping.