Pre-heat the oven to 425?F.
Set up three shallow dishes on the counter: Place about 1 cup of flour in the first dish. In the next dish, beat the eggs with the mustard and 2 tablespoons flour. In the last dish, combine the breadcrumbs with the grated cheese, salt and pepper.
Drizzle about 2 tablespoons EVOO on a baking sheet and place it in the oven until the EVOO is hot, about 5 minutes.
While the baking sheet is heating up, coat the veggies in the flour, then the egg mixture and lastly, the breadcrumb mixture, pressing to set the coating. Take the pan of hot EVOO out of the oven and coat the veggie fries in the EVOO. Roast them in the oven, tossing halfway through baking, until golden brown and crispy, about 15 minutes.