Preheat a large skillet with the EVOO, enough to go 1/4 inch up the sides of the skillet. Add the pretzel-coated chicken breast to the hot oil. Cook in a single layer, in two batches if necessary, about 3-4 minutes on each side, until cutlets' juices run clear and the breading is evenly browned. Transfer the finished chicken strips to the oven to keep warm. Repeat the same process with the zucchini.
While the zucchini is frying, heat a medium saucepot over medium heat, melt butter and add the flour. Cook flour and butter for 1 minute then whisk in the milk. When the milk comes to a bubble, stir in cheese and mustard with a wooden spoon. Season with salt and freshly ground black pepper, and remove from heat.
Transfer the fried pretzel-crusted chicken and zucchini pieces to a serving plate. Serve the mustard-cheddar sauce, carrot and celery sticks, and pickle spears alongside.