Mix the flax egg and set aside.
Combine the milk and butter in a microwave-safe bowl and heat for 1 minute. Set it aside for several minutes. Then add the flax egg and stir. The mixture should be between 105°F/40°C and 108°F/42°C (the same temperature as your finger if you dip it into the liquid).
In a large bowl, add the flour, sugar, yeast, and salt. Stir to combine. Pour this milk mixture into the flour mixture and stir together until a soft dough forms.
Knead the dough in the bowl with your hands or use a dough hook on a mixer for 5 minutes. Add 3 to 4 tablespoons of flour as needed to keep the dough from sticking to the bottom of the bowl. The dough should form into a ball but still be slightly sticky.
Spray the top with vegetable cooking spray, cover with a kitchen towel, and set it aside to rest, about 30 minutes.
Roll the Dough: Once the dough is ready, cover a working area with a sheet of plastic wrap. Place the dough on top of the plastic wrap and cover it with another sheet of plastic wrap. Use a rolling pin to roll the dough out to approximately 18x18 inches in size. It helps if you pull the top layer of plastic wrap off the dough and cover it again with the same sheet. It also helps to place an anti-slip pad to keep the dough from sliding as you roll it.
Add the Filling: Spread softened butter across the dough. Stir together the sugar and the cinnamon and sprinkle this over the butter, covering all of it except for about a half-inch of the long edge furthest away from you. Starting from the long end closest to you, roll the dough tightly until you're left with one long roll. Cut into 12 evenly sized rolls about 1 ½ inches wide.
Cut the Rolls: Prepare a baking dish by spraying a 9-inch round cake pan or pie pan with nonstick cooking spray. Place rolls in the prepared pan. Cover with a kitchen towel, and set aside to rise in a warm, draft-free area until doubled in size, about one hour. See post tips for the best place to rise cinnamon rolls.
When the rolls have doubled in size, preheat the oven to 350ᵒF/175°C. Place pan in a center rack and bake about 24 to 26 minutes until the rolls are golden in color.
While the rolls are baking, prepare the frosting. In a medium bowl, cream together the butter and vegan cream cheese until fluffy. Add the vanilla, followed by the powdered sugar beat until creamy. Stir in one tablespoon of the soy milk at a time to get a spreadable consistency.
When the rolls are ready, remove them from the oven and add frosting immediately. Serve warm.
Store cinnamon rolls in a sealed container at room temperature for up to 3 days or in the fridge for up to 7 days. These can be frozen for up to 2 months. To serve, cook in the microwave until heated through.