Slice head of cauliflower lengthwise to get a thin slab, a.k.a. steak. Heat 1 Tbsp. oil in large skillet; add cauliflower and season with salt and pepper. When seared on both sides, remove from pan.
Heat remaining 2 Tbsp. oil in skillet. Add mushrooms and saute until browned; season with salt and pepper.
Add shallot rings to remaining oil in pan and cook until golden brown.
Arrange plate with cauliflower steak at bottom, topped with sauteed mushrooms, chopped radicchio, shallot rings, and fresh thyme garnish.