Melt 2 tablespoons of the butter in a large skillet over medium-high heat. Swirl the butter around the pan to coat the bottom.
Open the chicken thighs so they lie relatively flat, then season both sides with salt and black pepper.
Arrange the chicken in the hot skillet without overcrowding the pan. Cook for 4 to 5 minutes per side, or until nicely browned and the chicken releases easily from the skillet.
Transfer the browned chicken to a plate. Cover loosely to keep warm. The chicken does not need to be completely cooked at this stage because it will finish cooking in the sauce.
Reduce the heat to medium. Add the remaining tablespoon of butter and the sliced shallots.
Cook the shallots for 4 to 5 minutes, stirring occasionally, until softened and lightly golden.
Pour the chicken stock into the skillet and increase the heat to medium-high.
As the stock comes to a boil, scrape the bottom of the skillet with a wooden spoon or spatula to loosen the browned bits left from cooking the chicken.
Stir in the chopped rosemary. Cook for 4 to 5 minutes, stirring occasionally, until the liquid has reduced by approximately half.
Return the chicken thighs and any accumulated juices to the skillet.
Add the frozen broccoli and thin lemon wedges around the chicken.
Cover the skillet and reduce the heat to medium-low. Simmer for 10 to 12 minutes, or until the broccoli is crisp-tender and the chicken is fully cooked.
Check the thickest part of the chicken with an instant-read thermometer. It should reach at least 165°F (74°C).
Remove the skillet from the heat. Spoon some of the lemon-rosemary pan sauce over each chicken thigh and serve immediately.