Preheat your oven to 325°F (160°C).
Bake the pie crust for 10–12 minutes to partially bake it. Remove from the oven and set aside.
Make the pie filling: in a medium saucepan, whisk together the white sugar and cornstarch. Add the coconut milk and stir to combine. Dollop the butter on top (the butter will melt into the mixture as it cooks). set the saucepan over medium-high heat and cook, whisking often, until the mixture thickens, 5–10 minutes. remove from the heat and whisk in the vanilla. set aside.
Make the topping: in a small bowl, mix together the white sugar and cinnamon. set aside.
Pour the prepared filling into the pie crust, filling it three-quarters full. drizzle on the melted butter and evenly sprinkle with the cinnamon sugar. (If you have leftover filling, pour it into a container and pop in the fridge for a delicious pudding!)
Bake for 25 minutes. The pie will be bubbling and still very liquidy. Carefully remove from the oven and allow to cool on a rack at room temperature. Once cooled, move the pie to the fridge until set, about 3 hours, or overnight is best.