In a small saucepan over low heat, bring the milk and coconut oil to 110-115°F (if it gets too hot, let the temperature come down or it will kill the yeast.) Stir in the vanilla. In a large bowl with a wooden spoon or in the bowl of a stand-mixer with a dough hook attachment, mix the flour, sugar, yeast and salt together. Slowly stream in the warm milk and oil mixture. If using a stand mixer, run it on a medium-low setting for about 10 minutes. If doing it by hand, remove the dough from the bowl and on a lightly floured surface with floured hands, knead it for about 10 minutes (the longer the better!). Try not to add much extra flour as this dough is softer and stickier than, say bread dough, and adding more flour will take away its final softness. Add only enough to keep working the dough; keep your hands lightly oiled helps, too. Wash out the bowl and grease it with coconut oil. Form the dough into a ball, place it in the bowl and turn it over so both sides are oiled. Lightly cover and let it rise in a warm place for 1 1/2 hours or until about doubled in size.
Place the coconut oil for the filling into a microwavable bowl (size of a cereal bowl). Microwave for 15 seconds, check it and if it's not melted, give it a stir and place it back in the microwave for another 10 seconds. Mix in the brown sugar and cinnamon.
Line a 9 x 13" pan with parchment paper. Preheat the oven to 350°F.
On a lightly floured surface, roll the dough out into a rectangle about 20 x 14". Sprinkle the cinnamon sugar mixture over the dough evenly. With the long end closest to you, roll the dough away from you evenly. When finished, push the roll over a bit so the seam is facing down. Using a very sharp knife, cut the roll into 12 even pieces (I find cutting it in half then each half in half, then each of those into 3 helps keep things even). Place the rolls into the prepared pan cut side down. Cover lightly and place in a warm spot for 30-60 minutes.
Uncover the rolls, place them in the preheated oven and bake for 22-28 minutes or until golden on top and a toothpick inserted into a doughy part comes out clean. Remove from the oven, let cool for 10-15 minutes and spread with vanilla glaze or enjoy as is! Once cooled to room temperature, cover with plastic wrap or place in an airtight container.
For the Glaze:
In a medium-sized bowl, beat the vegan butter until creamy. Add the confectioner's sugar a half cup at a time until incorporated. Add in the vanilla and 2 tbsp of the plant-based milk. Beat until all of the lumps are gone. If it's too thick, add more plant-based milk a tbsp at a time until it reaches the desired thickness. Spread over the rolls.