Preheat oven to 375°F (190°C).
Pound chicken to even thickness and season with salt, pepper, and Old Bay.
Sear chicken in olive oil and butter for 3–4 minutes per side until golden; set aside in a baking dish.
Mix crab topping: In a bowl, gently combine crab meat, mayo, green onions, lemon juice, seasonings.
Top chicken with crab mixture, pressing gently to adhere.
Make cream sauce: Sauté shallots and garlic in butter, deglaze with wine, reduce by half (~2–3 min).
Add cream, simmer, then stir in parmesan, mustard, and seasoning. Thicken sauce slightly.
Pour sauce around (not over) the chicken. Cover and bake 15 minutes.
Uncover and bake another 5–10 minutes, until chicken reaches 165°F and crab is golden.
Rest for 5 minutes before serving. Garnish with parsley and lemon wedges.