Combine all ingredients for the rolls (all-purpose flour through kosher salt) in the bowl of a stand mixer and mix on low to combine. Increase speed to medium for 2 minutes until a smooth dough ball forms. The dough will be slightly sticky. If needed, add more flour or non-dairy milk (1 Tablespoon at a time) to reach your desired consistency. Transfer the dough to a large, greased bowl. Cover with plastic wrap and let rise for 60-90 minutes, or until doubled in size.
Turn the dough onto a floured surface and roll into a 9x14 inch rectangle. Spread with the softened butter, the sprinkle with brown sugar and cinnamon, leaving ¼ inch of dough on the short end of the rectangle without any toppings (to make sealing easier).
From the short end of the rectangle, tightly roll the dough into a log. Press the end seam into the dough to seal shut. Using a sharp knife or dental floss, cut the dough into 8-12 rolls**. If using floss, wrap the string around the log, then pull tight to pinch and cut the dough.
Transfer the rolls to a greased 8x11-inch pan, face up. There should be some room for the rolls to expand within the dish. Cover with plastic wrap and let rise 30-45 minutes until they have expanded to fill the dish.
While the rolls rise, preheat your oven to 350°F. Bake for 25-30 minutes, until golden brown.
While the rolls cool, make the frosting. In a large bowl, whip the softened butter until light and fluffy, then add vanilla extract and a pinch of salt. With the mixer on low, add the confectioner's sugar and 1 Tablespoon milk. Mix until smooth. Add additional milk to reach your desired consistency, if needed. Smooth the frosting on the rolls and serve immediately.