Prepare dough: Place almond milk, melted vegan margarine, 1/2 cup sugar, salt, vanilla, and flour into a bread machine in the order listed. Start the Dough setting.
Mix warm water, 1 tablespoon sugar, and yeast in a glass measuring cup.
Preheat the oven to 300 degrees F (150 degrees C) for 1 minute. Turn the oven off and immediately place the yeast mixture inside to proof for 15 minutes.
Add proofed yeast mixture to the bread machine and let the cycle finish, about 1 hour 45 more minutes.
Remove dough and roll into an 12x18-inch rectangle. Mix brown sugar, walnuts, and cinnamon for filling in a bowl; spread evenly over the dough. Starting at the long end of the rectangle, roll dough into a log; slice log into 1-inch wide pieces. Transfer to a greased baking pan with cut-sides up.
Preheat the oven to 400 degrees F (200 degrees C) for 1 minute. Turn the oven off; immediately place the rolls inside along with an uncovered bowl of boiling water. Allow rolls to proof for 1 hour.
Remove rolls from the oven. Preheat to 350 degrees F (175 degrees C).
Bake rolls in the preheated oven for 15 minutes.
Meanwhile, mix powdered sugar, almond milk, and melted vegan margarine for glaze in a bowl.
Remove rolls from the oven and spread glaze evenly over top.