Stir the warm almond milk, yeast, and 1 tablespoon of the sugar together in a large bowl. Let it rest for 5 to 10 minutes until foamy.
Add the remaining sugar, melted vegan butter, egg, flour, and salt. Knead for about 8 minutes until the dough is smooth and elastic.
Place the dough in a greased bowl, cover, and let it rise in a warm spot for 1 hour, until doubled in size.
Roll the dough into a 14x10-inch rectangle. Spread the softened vegan butter over the surface, then sprinkle evenly with the brown sugar and cinnamon.
Roll the dough tightly from the long side into a log. Slide a piece of unflavored dental floss under the log and pull it up to slice 12 even rolls, then arrange them in a greased 9x13-inch pan.
Cover and let the rolls rise for 30 to 40 minutes until puffy. Preheat the oven to 350°F during the last 10 minutes.
Bake for 22 to 25 minutes, until golden on top and cooked through in the center.
Whisk the powdered sugar, almond milk, and vanilla into a smooth icing and spread it over the warm rolls so it melts into every swirl.