Trim ends off green beans. Zest and halve lemon. Cut one half into two wedges and juice the other half. Peel potatoes, halve lengthwise, and cut into ½" half-moons. Mince parsley, both stems and leaves. Coarsely crush or chop pine nuts. Rinse chicken breasts, pat dry, and season both sides with ¼ tsp. salt and ¼ tsp. pepper.
Roast the Potatoes
Add potatoes, 1 Tbsp. olive oil, ½ tsp. salt, and ¼ tsp. pepper to prepared baking sheet. Toss to coat and spread into a single layer. Roast until golden brown and tender, 12 minutes. Potatoes will finish cooking in a later step. Carefully, move potatoes to one side to make room for chicken. While potatoes roast, bread chicken.
Bread the Chicken
Add liquid egg to a medium mixing bowl and panko breadcrumbs and pine nuts to a second medium mixing bowl. Dip chicken breasts in liquid egg, coating completely. Shake off excess and dip chicken breasts in pine nut-panko mixture. Any extra pine nut-panko mixture should be pressed into chicken afterward.
Cook the Chicken
Heat a medium non-stick pan over medium heat and add 2 Tbsp. olive oil. Add chicken breasts to hot pan and cook until browned, 2-3 minutes per side. Transfer to baking sheet with potatoes, place on open side, and sprinkle ¼ tsp. salt. Roast until browned and chicken reaches a minimum internal temperature of 165 degrees, 4-5 minutes. Sprinkle potatoes with 1 tsp. lemon zest, 1 tsp. lemon juice, feta cheese, and parsley (reserve a pinch for garnish). Wipe pan clean and reserve.
Cook the Green Beans
Return pan used to cook chicken to medium-high heat, add 1 tsp. olive oil, green beans, ¼ tsp. salt, and ¼ tsp. pepper to hot pan. Cook, stirring often, until lightly charred, 2 minutes. Add ¼ cup water and cook, stirring often, until bright green and crisp-tender, 2-4 minutes.
Finish the Dish
Sprinkle remaining parsley on chicken breasts. Set green beans, Greek potatoes, and chicken breast on a plate and squeeze lemon wedge over to taste.