Preheat oven to 400°F. Arrange oven racks so a large skillet and baking sheet can fit at the same time.
Melt 3 tablespoons butter in a saucepot over medium heat. When butter foams, whisk in 3 tablespoons flour, cook 1 minute then whisk in stock and warm half-and-half. Let thicken and season with salt, pepper and nutmeg to taste. Stir in cheeses and keep warm over low heat.
Preheat a large, cast-iron skillet over medium-high heat.
Cut chicken away from bones, keeping the skin intact. Place chicken between plastic wrap or parchment and pound to 1/2-inch thick. Season on both sides liberally with salt and pepper. Dredge very lightly in flour, shaking off all excess.
Arrange broccolini on a baking sheet and toss with 2 tablespoons or so olive oil, garlic, salt and pepper. Roast 15-18 minutes until crispy at edges of broccolini tops.
Add olive oil, 1 turn of the pan, to skillet and melt remaining tablespoon butter into oil. When it foams, add chicken skin-side down; press down on chicken as it cooks with spatula to make skin extra crispy. Cook 5-6 minutes then transfer to oven for 5 minutes more. Remove, turn the chicken and let stand a minute or two. Transfer to plates and douse with lemon juice.
Plate the chicken with roast broccolini alongside. Ladle the Mornay sauce over the top of broccolini and chicken. Garnish with crispy onions and serve.