Combine 2 cups water, the cauliflower, whole rosemary sprig and 1/2 teaspoon salt in a pot and bring to a boil. Cover, reduce the heat to low and cook until the cauliflower is very tender, about 10 minutes. Drain and transfer to a food processor, discarding the rosemary. Add the cheese and 1 tablespoon butter and puree. Transfer to a bowl and fold in the chives; season with salt and pepper. Cover to keep warm.
Season the steaks on both sides with salt and pepper. Melt 1 tablespoon butter in a large skillet over high heat until foamy and starting to brown, about 1 minute. Add the steaks and cook until browned, about 1 1/2 minutes per side. Transfer to a plate.
Add 1 tablespoon butter to the skillet. Reduce the heat to medium high and add the shallot, garlic and chopped rosemary. Cook, stirring, 30 seconds. Add the mushrooms and cook until tender, about 3 minutes. Add the wine; cook until reduced by half, about 2 minutes. Add the chicken broth; cook until thickened, about 4 minutes. Add the steaks and any accumulated juices and the remaining 1 tablespoon butter to the skillet. Cook, turning the steaks to coat, 1 minute. Serve with the cauliflower puree.