Take the vegan butter for the cake and cinnamon swirl out of the fridge to soften.
Line the bottom of a round 9-inch cake pan with parchment paper and grease the sides to prevent sticking. Preheat the oven to 350°F/175°C.
Make the vegan buttermilk. Combine the 3/4 cup + 2 tablespoons non-dairy milk and apple cider vinegar and stir. Set aside to curdle.
Prepare the aquafaba. Pour 1/3 cup of the liquid from a can of chickpeas into a medium bowl. You can also use the liquid from chickpeas cooked from scratch. With an electric mixer, beat the aquafaba on medium speed until it starts to get foamy, about 45-60 seconds.
Whisk together the cake flour, salt, baking powder, and baking soda in a medium bowl.
In a large bowl, add the softened 1/3 cup vegan vegan butter, cane sugar, and oil until well incorporated and fluffy, 45 seconds or so. Add in the whipped aquafaba and vanilla extract, and mix until just combined, 15-20 seconds .
Add half of the flour mixture into the butter-sugar mixture, beating on low speed until just combined. Then add the vegan buttermilk and beat until just combined. Finally, add in the remaining flour mixture until you have a mostly smooth batter, but do not overmix (it’s okay if there are a few flour pockets remaining) to prevent the cake from getting dense. The batter should be light and fluffy. Transfer the batter to the prepared 9-inch cake pan.
Make the cinnamon swirl filling: Add the 1/2 cup butter, brown sugar, cinnamon, and 1 tablespoon flour to a medium bowl. Using an electric mixer on medium speed, cream together until evenly combined.
Scoop the cinnamon swirl filling into a piping bag or a ziplock bag with the tip cut off. Starting in the center, pipe the filling into a concentric circle to mimic the look of a cinnamon roll. Alternatively, you can simply dollop the cinnamon filling evenly over the cake better and use a skewer or knife to marble and swirl over the cake.
Bake the cake for 40-45 minutes, or until a toothpick inserted in center comes out mostly clean, and it feels firm to the touch and the edges are lightly browned. My cake was done in exactly 40 minutes.
Remove the cake from the oven and allow to cool on a cooling rack for 10-15 minutes before removing from the cake pan and serving.
Make the sugar icing while the cake is resting. Add the powdered sugar, 2 tablespoons nondairy milk, and vanilla to a medium bowl. Whisk until smooth, but still thick and viscous. Add more milk as needed. Drizzle the icing over the warm cake and serve warm.