Preheat the oven to 425°F (220°C). Line two baking sheets with parchment paper or lightly grease them. Cut carrots, zucchini, red pepper into uniform sticks about 3 inches long and 1/2 inch thick. Break broccoli into small, bite-sized florets. Pat vegetables dry with paper towels so coating adheres.

Prepare coating stations

Place the flour in a shallow bowl and season it with half the salt and pepper. In a second bowl, beat the eggs until smooth. In a third bowl, combine panko breadcrumbs, grated cheddar, garlic powder, paprika, remaining salt and pepper, and grated Parmesan if using. Stir to combine.

Coat the vegetables

One or two pieces at a time, toss vegetable sticks lightly in the seasoned flour to coat and shake off excess. Dip into the beaten eggs, letting excess drip off, then press into the breadcrumb-cheese mixture, coating all sides. Place coated pieces on the prepared baking sheet. For broccoli, press crumbs into the florets so they adhere.

Bake until crisp

Drizzle or lightly brush the coated vegetables with the olive oil (or spray with cooking spray) to help browning. Spread them in a single layer leaving small gaps for air circulation. Bake for 12–16 minutes, turning once halfway through, until coating is golden and vegetables are tender-crisp. Smaller pieces like broccoli may cook faster—remove when golden and tender.

Finish

If using grated Parmesan or extra cheese, sprinkle over hot dippers as they come out of the oven so it melts slightly. Let cool for 2–3 minutes before serving so coatings set and they aren’t too hot for kids.

Creamy Yogurt Dip

Mix Greek yogurt, mayonnaise (if using), honey, lemon juice, chopped dill, salt and pepper in a small bowl until smooth. Taste and adjust sweetness or acidity to suit kids (more honey if they prefer sweeter, more lemon to brighten). Chill briefly while dippers bake.

Serve and Optional Finishes

Arrange warm veggie dippers on a platter with the yogurt dip in a small bowl. Offer ketchup or extra grated cheese on the side for picky eaters. Sprinkle chopped chives on the dip or dippers for color. Encourage kids to dunk and try each vegetable—cut some pieces into smaller sizes if serving to toddlers.