Chop cauliflower stems into small pieces. Measure out 1/2 cup of cauliflower florets and chop into small pieces. Set aside.
Bring a large pot of lightly salted water to a boil. Add cauliflower florets and cook uncovered until crisp tender, 4 to 5 minutes. Drain. Toss with Parmesan cheese and chives.
Preheat oven to 425 degrees F (220 degrees C).
Melt butter in a skillet over low heat. Add reserved stems and chopped florets, shallot, and garlic; cook and stir until tender but not brown, 3 to 5 minutes. Pour in vegetable broth and bring to a boil. Cook over high heat until broth evaporates completely, 8 to 10 minutes. Remove from heat.
Pour 1 cup heavy cream into the skillet and puree mixture with an immersion blender until smooth. Stir in horseradish, salt, and black pepper. Transfer to a baking dish and add blanched florets; toss to coat evenly with cream mixture. Drizzle with 2 tablespoons heavy cream and top with Gruyere cheese.
Bake in the preheated oven until golden brown, 20 to 30 minutes.