Peel and halve shallot. Slice thinly, then mince a few slices, resulting in 1 Tbsp. minced shallot. Add sliced shallots to a medium mixing bowl. Halve lemon and squeeze one half into bowl with shallots. Quarter other lemon half. Add ¼ tsp. salt and ¼ tsp. pepper to bowl, toss, and set aside to pickle, 15 minutes. While shallot pickles, prepare ingredients.
Prepare Ingredients and Roast Potatoes
Stem and mince parsley. Halve fingerling potatoes lengthwise. Toss potatoes with 1 Tbsp. olive oil, ½ tsp. salt and ¼ tsp. pepper on prepared baking sheet. Roast until crisp and golden, 18-20 minutes. Remove from oven and toss with everything bagel seasoning and half the parsley (reserve remaining for sauce).
Bread the Chicken
Rinse chicken breasts and pat dry. Add liquid egg to another medium mixing bowl and breading to a third mixing bowl. Place chicken into liquid egg, add ¼ tsp. salt and ¼ tsp. pepper, and toss to coat. Transfer chicken breasts to breading bowl and toss to coat completely.
Cook the Chicken
Line a plate with a paper towel. Place a medium pan over medium heat. Add 3 Tbsp. olive oil to hot pan and let oil heat up. Add chicken breasts to hot oil and sear until browned and chicken reaches a minimum internal temperature of 165 degrees, 3-5 minutes per side. Remove to paper-towel lined plate, wipe pan clean, and reserve.
Make the Sauce
Return pan used to fry chicken to medium heat. Add 2 tsp. olive oil and minced shallots and cook until aromatic, 30 seconds. Add wine and cook until reduced by half, 1-2 minutes. Remove from burner, swirl in butter, and add remaining parsley.
Plate the Dish
Place potatoes on a plate. Set chicken next to potatoes and place sauce in front of chicken.