Preheat oven to 425F.
Set up 3 shallow dishes on the counter: Place about a cup of flour in one dish. In the next dish, beat the eggs with mustard and 2 tablespoons flour. In the last dish, combine breadcrumbs with grated cheese, salt and pepper.
Drizzle about 2 tablespoons EVOO on a baking sheet and place into the oven until the oil is hot, about 5 minutes.
While the pan is heating up, coat the veggies in flour, then egg mixture, then breadcrumb mixture, pressing to set coating. Take the pan of hot oil out of the oven and coat the veggie fries in oil. Roast them in the oven, tossing halfway through baking, until golden brown and crispy, about 15 minutes.
Serve immediately.