Prep the cauliflower: Cut into small, even florets so they cook quickly and brown well.
Pat dry with a towel to remove excess moisture.
Brown the sausage: Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add sliced sausage in a single layer and cook 3 to 4 minutes per side until browned. Transfer to a plate, leaving the rendered fat in the pan.
Sear the cauliflower: Add remaining olive oil if the pan looks dry.
Add cauliflower in an even layer and let it sear undisturbed for 2 to 3 minutes to develop color. Stir and continue cooking 6 to 8 minutes until lightly charred and crisp-tender.
Season and steam-finish: Sprinkle in smoked paprika, red pepper flakes, thyme, salt, and pepper. Toss to coat.
If the cauliflower needs a little help softening, splash in 2 tablespoons water, cover, and steam 2 minutes. Uncover and let moisture cook off.
Add the garlic butter: Reduce heat to medium. Push cauliflower to the sides of the pan.
Add butter to the center; once melted and foamy, stir in minced garlic. Cook 30 to 45 seconds until fragrant, then toss everything to coat in the garlic butter.
Return the sausage: Add sausage back to the skillet and toss. Cook 1 to 2 minutes to warm through and mingle flavors.
Finish bright: Remove from heat.
Squeeze in lemon juice and sprinkle with parsley. Taste and adjust salt and pepper. Add Parmesan if you like.
Serve: Enjoy straight from the skillet, or spoon over rice, mashed potatoes, or creamy polenta.