Preheat oven to 350° F, and line muffin pan with paper liners.
In a large bowl, whisk together the dry ingredients: flour, corn starch, baking powder, baking soda, and salt.
In a smaller bowl, cream together the coconut oil and sugar until they’re really well combined. Then add the remaining wet ingredients: coconut milk, vinegar, and vanilla extract. Beat with a fork or an electric mixer until the wet ingredients are really well-combined.
Pour the wet ingredients into the bowl with the dry, and beat again, until well-combined. Some very small lumps are okay, but break up any bigger ones. The batter should be pretty thick, but if it's way too thick, add another 1 to 2 tablespoons coconut milk or other non-dairy milk, and mix in. Divide the batter between your lined muffin tins, filling each cup only halfway. Bake for 20 minutes, then pull them out to cool in the pan until they’re cool enough to handle.
Once you can touch the cupcakes without burning your fingers, transfer them to a plate, and stick them in the fridge for 20 minutes to chill (or until you’re ready to frost).
While they bake, make the coconut frosting
In a large bowl, use a handheld or stand mixer to beat together the vegan cream cheese and butter until they're well combined.
Add the confectioner's sugar and salt, and beat on low until the sugar is mostly combined into the cream cheese-butter mixture. Then, turn the mixer up to high and beat for another couple of minutes, until it's light and fluffy. Add more confectioner's sugar by the tablespoon, if needed, to reach your desired consistency. While you're beating on low, it's going to seem like it's too dry, but don't add any liquid until after beating on high for a couple of minutes. The texture changes dramatically!
Add the coconut and beat on low, just until it's combined.
Assemble the coconut cupcakes
Spread or pipe the coconut frosting onto the cupcakes. Sprinkle with the shredded coconut, then chill in the fridge until you’re ready to serve.
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Notes
You can make these a few hours ahead or even the day before serving, but you want to frost just before serving. If you’re going to store them overnight, keep them in a covered container in the fridge.Unfrosted, they'll keep in the fridge for up to five days. With frosting, they'll keep for two to three days.