Put the flour in a wide, shallow dish and season generously with salt and pepper -- stir through with a fork.
Put the breadcrumbs in a second wide, shallow dish, add the parmesan and oregano, and season generously with salt and pepper. Stir through until everything is well-combined.
Crack the eggs into a third dish and beat them lightly with the fork.
Line a baking sheet with a double layer of paper towels and get out a large, clean plate. Coat each of the chicken strips with the seasoned flour, shaking off any excess. Dip them in the egg mixture, and then in the breadcrumbs, pressing lightly so that the breadcrumbs adhere. As you finish each one, set it on the plate.
Heat 2 tablespoons of oil in a large, heavy skillet (I like to use cast iron) over medium heat. Add 1 tablespoon of the butter. When the butter has finished foaming and just starts to brown, gently lay half of the chicken strips in the pan, being careful not to crowd them. Cook for 1 1/2 to 2 minutes, until the bottom is golden brown and crispy. Use tongs to turn them over and cook for another minute or two, until both sides are evenly browned and the chicken is cooked through. Transfer the chicken fingers to the paper-towel lined baking sheet and repeat with the rest of the chicken, adding more oil and butter to the pan as needed. Serve the chicken fingers warm. (It reheats well in a 300° F oven the next day.)
Editors' Note: Alternatively, you can bake these chicken fingers in the oven, as Amanda Hesser suggests in A New Way to Dinner. Heat the oven to 450° F. Generously coat the base of a rimmed baking sheet with vegetable oil and add 2 tablespoons of butter. Place the baking sheets in the oven to heat. When the butter has finished foaming and just starts to brown, gently lay the chicken strips on the baking sheets. Cook for 10 minutes, until the bottoms are golden brown and crispy. Use tongs to turn them over and cook for another 10 minutes, until both sides are evenly brown and the chicken is cooked through. Add more butter and oil as needed.