- Heat 2 teaspoons oil in 10-inch ovensafe skillet over medium-high heat until just smoking. Add broccoli and cook until spotty brown, about 2 minutes. Add ¼ cup broth, cover, and cook until just tender, about 2 minutes. Transfer broccoli to plate and wipe out skillet with paper towels.
- Place 2 tablespoons flour in shallow dish. Pat chicken dry with paper towels and season with salt and pepper. Dredge chicken in flour, shaking to remove excess. Heat remaining 1 tablespoon oil in now-empty skillet over medium-high heat until just smoking. Cook chicken until golden brown, about 3 minutes per side. Transfer chicken to plate.
- Reduce heat to medium, add butter and shallot to now-empty skillet, and cook until fragrant, about 30 seconds. Stir in remaining 2 teaspoons flour and cook for 1 minute. Whisk in remaining 1 cup broth, cream, sherry, and Worcestershire and bring to simmer, scraping up any browned bits. Return chicken to skillet and simmer until it registers 160 degrees, 8 to 10 minutes, flipping halfway through cooking.
- Adjust oven rack 9 inches from broiler element and heat broiler. Transfer chicken to cutting board and continue to simmer sauce until reduced to ½ cup, 5 to 7 minutes. Off heat, whisk in ¼ cup Parmesan and lemon juice. Season sauce with salt and pepper to taste. Cut chicken into ½-inch-thick slices and stir chicken and broccoli into sauce in skillet. Sprinkle with remaining ½ cup Parmesan and broil until spotty brown, about 5 minutes. Serve.
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