First, pour the nondairy milk in a microwave-safe bowl, and heat for 30-45 seconds. You want the milk to be warm, but not burning hot... a little warmer than body temperature is perfect.
To the warm milk, add the oil, sugar, and yeast. If you are using active dry yeast, set this mixture aside for 10-15 minutes to allow the yeast to bloom. It is ready once it starts to get foamy and bubbly on the top. If you are using instant yeast, you do not have to wait and can go directly to the next step.
In a large bowl, mix the wet ingredients with the 2 cups of flour until a soft dough forms. Now it is time for the first rise. Cover your bowl and put it somewhere warm for 1 hour.
After the first rise, add in the salt, baking powder, baking soda, and additional flour. Knead until all the flour has been absorbed, and add more if needed. You want the dough to be soft and smooth but not too sticky!
Now it is time for the second rise. Cover the bowl with a tea towel again, and let rise for an additional 3+ hours. I like to let it rise overnight at this point.
After the second rise, it is time to shape your yeast dough.
Lightly flour a clean surface, and dump the dough on top. Knead it for 2-3 minutes, adding more flour as needed. The dough should be smooth and easy to work with, so add flour if it's too sticky!
Using a rolling pin, roll the dough out into a rough rectangle, making sure you don't roll it out too thin (somewhere between ½-¼ inch).
To prepare the filling, combine the softened vegan butter with the brown sugar, cinnamon, and vanilla. Mix until smooth and creamy, then spread all over the surface of your prepared dough, all the way to the edges. Starting on the long side closest to you, roll up the dough.
Using a large, smooth knife, cut 1½-2 inch slices of dough, and place them in a greased 8x8 pan.
At this point, cover the pan and let it rise one more time for 30 minutes.
When ready to bake, preheat the oven to 375F. Bake the rolls 20-25 minutes, until the tops are golden brown and no longer doughy.
While the rolls cook, make the glaze. In a small bowl, whisk together the nondairy milk, vanilla, and powdered sugar until thick and smooth. Make it thicker than you think, as the glaze will melt slightly when poured over the warm cinnamon rolls.
Let rolls cool slightly before glazing, and enjoy!