In a large bowl, whisk flour, instant yeast, sugar, and salt. Add warm milk, melted vegan butter, and flax egg. Stir to form a shaggy dough. Knead on a floured surface 8 to 10 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise 1 hour until doubled.

Punch down dough. Roll into a 12x16-inch rectangle. Spread with soft vegan butter, leaving a border. Mix brown sugar and cinnamon; sprinkle over butter. Roll tightly from a long side into a log.

Slice log into 9 rolls and arrange in a greased 9x9-inch pan. Cover and let rise until puffy, 30 to 40 minutes. Meanwhile, preheat oven to 350°F (175°C).

Bake 22 to 25 minutes until golden. Cool 5 minutes in pan.

Whisk powdered sugar, 2 tbsp non-dairy milk, and vanilla until smooth. Add more milk for thinner glaze. Drizzle over warm rolls and serve.