Move oven rack to bottom third of oven and heat the broiler.
Heat 1 Tsps vegetable oil in a large skillet over medium high heat. Add broccoli and cook 1-2 minutes until brown spots start to appear. Add ½ cup of chicken broth, cover and steam broccoli another 2 minutes.
Transfer the broccoli to a platter.
Dry the skillet. Add remaining 2 Tbsps vegetabel oil to the skillet. While it is heating up, season the chicken breasts with salt and pepper and dredge in the flour to coat. Cook the chicken 2-3 minutes per side until lightly browned. Transfer breasts to a separate platter.
Remove the skillet from the heat and add the shallots. After a minue add the remaining 2 cups of chicken broth and the whipping cream. Return the skillet to the heat and scrape up any browned bits from the bottom of the pan. Add chicken and any reserved juices back to the skillet and simmer about 10 minutes until chkcen is cooked through. Remove breasts to a clean plate. Continue simmering the sauce until has reduced to 1 cup. This may take up to 10 minutes. Add the sherry and Worcestershire sauce and simmer another 3 minutes.
Whisk the egg yolks an dlemon juice in a small bowl. Stir in 1 cup of the cheese into the cream sauce in the skillet. Whisk ¼ cup of the cream sauce in to the bowl of egg yolks. Stiry vigorously while adding so you don't cook the egg yolks.
Remove the skillet from the heat and whisk the egg mixture into the skillet. Whisk in the butter after the eggs have been incorporated.
Cut the chicken cross=wise into ½" slices and arrange in a broiler proof pan. Scatter the broccoli over the chicken. Pour the sauce over the chicken and broccoli. Top with the remaining ½ cup of cheese. Broil 3-5 minutes until golden brown. Serve immediately.
Serve immediately with noodles or baked potatoes.