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 1/2" half-moons. Mince parsley, both stems and leaves. Coarsely crush or chop pine nuts. Rinse chicken breasts, pat dry, and season both sides with 1/4 tsp. salt and 1/4 tsp. pepper.

Add potatoes, 1 Tbsp. olive oil, 1/2 tsp. salt, and 1/4 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.

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.

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 1/4 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.

Return pan used to cook chicken to medium-high heat, add 1 tsp. olive oil, green beans, 1/4 tsp. salt, and 1/4 tsp. pepper to hot pan. Cook, stirring often, until lightly charred, 2 minutes. Add 1/4 cup water and cook, stirring often, until bright green and crisp-tender, 2-4 minutes.

Sprinkle remaining parsley on chicken breasts. Set green beans, Greek potatoes, and chicken breast on a plate and squeeze lemon wedge over to taste.