12 hours or so before you wish to bake, prepare the dough.
Using a food processor or a pastry cutter, combine butter and flour until the mixture looks sandy and uniform. If using a food processor, empty contents into a large bowl. Add starter, sugar, and milk and very gently mix until dough only just comes together. It is important not to overmix at this stage. Cover the dough with plastic wrap or a clean, damp towel and let rest at room temperature 12-18 hours.
In a small bowl, mix salt, baking powder, and baking soda. Sprinkle over the rested dough and mix with your hands until the ingredients are incorporated. Dough should slacken considerably and feel very tender and light. Again, be careful not to overmix.
Preheat oven to 400°F.
Lightly flour a clean level surface and roll out the dough until it is roughly 1/4" thick, and in a rectangular shape.
In a medium bowl, mash warm butter, sugar, cinnamon, and ginger with a fork until a paste forms. Spread the paste evenly over the dough using the back of a spoon or a spatula.
Roll the dough up lengthwise as tightly as possible. Cut the ends off of the log, then cut the remaining dough into roughly 1" thick rounds.
Place buns in a buttered cast iron skillet, cookie sheet, or muffin tin and bake 20-25 minutes, or until buns are golden brown at the edges.
Meanwhile, prepare the icing in a medium bowl. Combine room temperature cream cheese, vanilla, salt, and milk with a whisk or spatula. Gradually add powdered sugar until incorporated, adding more sugar as desired.
Drizzle buns with icing and serve immediately. Keeps in the fridge up to 3 days. Reheat in small bursts in the microwave for delicious leftovers.