Combine warm soy milk, granulated sugar, and yeast in a bowl; let stand for 5 minutes until frothy.
Whisk in 1/3 cup melted vegan butter and salt. Gradually incorporate flour until a soft dough forms.
Knead the dough on a lightly floured surface for 7 minutes until smooth and elastic.
Place dough in a greased bowl, cover, and allow to rise in a warm environment for 60 minutes or until doubled in volume.
Mix brown sugar and cinnamon in a shallow bowl. Place 1/2 cup melted butter in a separate bowl.
Deflate dough and divide into approximately 42 small spheres.
Dip each sphere into the melted butter and then roll in the cinnamon-sugar mixture until coated.
Arrange spheres in layers within a greased 10-inch Bundt pan. Cover and let rise for 45 minutes.
Preheat oven to 350°F (175°C). Bake for 30 to 35 minutes until the top is golden and bubbling.
Cool in the pan for 10 minutes before inverting onto a serving platter.
Whisk powdered sugar, 2 tablespoons soy milk, and vanilla extract; drizzle over the warm bread before serving.