Heat 2 tablespoons of butter in a large skillet over medium-high heat until melted and bubbly.
'Season each side of chicken thighs with salt and black pepper. Add seasoned chicken to the hot skillet and cook for 4-5 minutes per side, the chicken will be golden brown and will easily release from the bottom of the skillet. Remove from heat and transfer chicken to a plate. Cover and keep warm.
Reduce heat to medium and add remaining butter and sliced shallots to the skillet. Saute shallots until they have softened and are slightly golden. This will require 4 to 5 minutes and you may need to stir them occassionaly
Once the shallots are sauteed, add chicken stock to skillet and return heat to medium-high, Bring chicken stock to a boil. Once boiling use a wooden spoon or spatula to loosen brown bits from the bottom of the pan. You'll deglaze the pan.
Add chopped rosemary leaves and continue cooking until stock is reduced by half. This will take another 4-5 minutes, stir occassionally,
Return chicken and any juices to the skillet. Also, add the frozen broccoli and thin lemon wedges. Cover and reduce heat to medium-low, and simmer for 10-12 minutes, or until broccoli is tender (not mushy) and the chicken is cooked. (Test with an instant-read thermometer. The internal temperature of the chicken should read 165°F. Note: If the broccoli cooks sooner than the chicken, remove to a serving plate and cover to keep warm.
Remove from heat and serve immediately. Place chicken in a bowl and then spoon pan sauce over each chicken thigh. Garnish with lemon wedges, and Enjoy.