Preheat the oven to 500°F (if using a baking sheet) or as high as it will go (if using a baking stone/steel; place the stone/steel in the oven before you start preheating).
Melt the butter in a skillet over medium heat. Add the garlic and mushrooms and cook, stirring frequently, until the mushrooms are just tender. Sprinkle with salt and pepper to taste and remove the pan from the heat.
Stretch or roll out your dough to a 12-inch circle, then transfer it to an oiled baking sheet or a floured pizza peel (if using a baking stone/steel).
Brush the dough lightly with olive oil, making sure to get all the way to the edges. Scatter the mushrooms evenly on top, followed by the spinach. Season with a pinch of salt and a grind or two of black pepper, then add the Parmesan cheese and sprinkle with paprika. (Make sure you don't add clumps of paprika; hold the jar about 6 inches above the pizza and sprinkle as evenly as you can in a zig-zag pattern.) Drizzle with the cream.
Transfer the pizza to the oven and bake until the crust is golden and the cheese has browned in spots, 8 to 10 minutes on the baking sheet, 6 to 8 minutes on the baking stone/steel.
Remove the pizza from the oven and sprinkle with the chives. Wait about 5 minutes before slicing and serving.