Quickly blanch the broccoli in boiling water for 3 minutes until crisp-tender. Drain and transfer to an ice bath to stop the cooking. Drain and set aside.
In the meantime, lay the sausages in a skillet with ½ cup water. Cook half covered for 10 minutes on medium heat, turning regularly. When the water has evaporated, add 2 tablespoons butter, half the minced garlic, and red chili pepper flakes and continue cooking for a couple of minutes, until sausages have browned in the garlic butter sauce. Transfer sausages to a plate and set aside.
In the same skillet, lower the heat and melt the remaining tablespoon butter. Add chopped parsley, thyme, remaining garlic, Italian seasoning, red crushed chili pepper flakes, and blanched broccoli florets and cook for 4 to 5 minutes, stirring regularly, until cooked to your liking. Deglaze with lemon juice, vegetable or beef stock, hot sauce, and reduce the sauce for a couple of minutes.
Add the garlic butter sausages back to the pan with the broccoli and reheat quickly. Spoon the sauce over the sausage and broccoli. Adjust seasoning with pepper and serve the garlic butter sausages and broccoli skillet immediately, garnished with more crushed chili pepper, fresh herbs, and a slice of lemon if you like.
