Whisk warm non-dairy milk, yeast, and a teaspoon of the sugar in a large bowl. Rest for 5 minutes until lightly foamy.
Stir in the remaining sugar, melted vegan butter, vanilla, and salt. Add flour gradually until a soft dough forms.
Knead the dough on a lightly floured surface for 5 to 7 minutes, adding only enough flour to keep it from sticking.
Place the dough in a lightly greased bowl, cover, and let it rise in a warm spot until noticeably puffy, about 45 minutes.
Roll the dough into a rectangle about 1/4 inch thick. Brush with melted vegan butter, then sprinkle evenly with brown sugar and cinnamon.
Roll the dough tightly from the long side and cut it into 14 even pieces with a sharp knife or unflavored dental floss.
Arrange the rolls cut-side down in a greased baking dish, cover with a towel, and let them rise for 30 minutes.
Heat the oven to 350°F. Brush the rolls with a little melted vegan butter and bake for about 20 minutes, until lightly golden.
Whisk powdered sugar, maple syrup, and non-dairy milk into a smooth glaze. Cool the rolls for 5 minutes, drizzle, and serve warm.