Preheat the oven to 180°C and grease and line the bottoms of two 7-inch round cake pans with baking paper.
Make the cake. In a medium bowl, whisk together flour, baking powder, bicarbonate of soda and salt. Set aside.
In a large bowl using an electric mixer with a paddle attachment, beat the dairy-free butter, vanilla and sugar together. Add in the egg and whites and beat until smooth and creamy. Scrape down the edges of the bowl with a rubber spatula. Add in half the flour mixture followed by half the milk, then repeat, beating to fully combine everything in between each addition. Add in sprinkles and mix through slowly with a spatula.
Divide the batter evenly between the two pans and place in the oven. Bake for 20-25 minutes or until the tops are lightly springy when touched and a skewer when inserted comes out clean.
Let the cakes cool in the pan for 10 minutes before turning out onto a wire rack to cool completely.
Make the frosting. Using a hand mixer or a stand mixer with the paddle attachment fitted, beat dairy-free butter and vanilla for 1-2 minutes until creamy. Add in 3 cups of icing sugar and beat on medium until combined. As mixture thickens, add in dairy-free milk a little at a time. If frosting is too thin, add in an extra ½-1 cup of icing sugar until it reaches the right consistency. Add a pinch of salt to taste if the frosting is too sweet.
Assemble the cake. Add a little bit of frosting to a cake plate and set down the first cake. Add a generous amount of frosting followed by the second cake upside down so the top is flat. Using a cake knife, frost the top and sides with the remaining frosting. With any leftovers, add to a piping bag and pipe swirls around the top. Decorate with more rainbow sprinkles by pressing them around the bottom and sprinkling over the top.