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 bottom third of the oven before you start preheating). Let the oven preheat for at least 30 minutes. Then, if you’re using a baking stone or steel, switch the oven to Broil on high.
Melt the butter in a cast iron skillet over medium heat. Add the shallot and sauté just until it has softened, then add the mushrooms and cook for about 5 minutes, stirring frequently, until they are tender and have released their liquid. Remove the skillet from the heat and season the mushrooms with salt and pepper to taste.
Stretch or roll out your dough to a 12- to 14-inch circle, then transfer it to an oiled or parchment-lined baking sheet or a lightly 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. Using a slotted spoon, transfer the mushrooms to the dough and spread them out, then scatter on the fennel and sprinkle with the Parmesan cheese. Drizzle the cream evenly over top.
Transfer the pizza to the oven and bake/broil 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 let it rest for 5 minutes, then sprinkle on the tarragon and finish with a pinch of salt and a grind or two of black pepper. Slice and serve.