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).
Place the chives and spring onions in separate bowls and add 1 tablespoon of olive oil to each. Toss to coat, then season the chives and spring onions with a pinch of salt and a grind of pepper; toss again.
If you're using a baking stone/steel, turn the oven to Broil on high.
Stretch or roll out your dough to a 12-inch circle, then transfer it to an oiled baking sheet or a lightly floured pizza peel (if using a baking stone/steel).
Drizzle the dough very lightly with olive oil, and use a pastry brush to spread the oil all the way to the edges of the dough. Scatter on the spring onions, then the chives. Drizzle half of the cream evenly over top.
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 immediately drizzle the remaining cream evenly over top and season with a pinch of salt. Let the pizza sit for 5 minutes or until the cream has congealed, then slice and serve.