WHISK egg with milk in a large bowl until combined. Stir in bread until soft. Mash in salmon, crushing any bones. Stir in dill, chives and Dijon. Season with fresh pepper.
FORM 1/4-cup packed portions of salmon mixture into patties about 1/2 in. thick. Heat a large non-stick frying pan over medium-high. Add 3 tsp oil, then patties. Cook until golden, 2 to 3 min per side. Transfer cakes to a plate.
WIPE pan clean, then return to heat over medium. Add remaining 1 tsp oil, then green beans and salt. Cook, stirring often, until tender-crisp, about 5 min. Serve with salmon cakes and sour cream.