To roast your beet, wrap whole beet in aluminum foil and cook in 400°F oven until tender, about 1 hour. Once cool to the touch, you can cut away skin and top with sharp paring knife. Reserve any juice for the icing.
Combine cooled and peeled roasted beet and 1 cup buttermilk in blender and puree until smooth. Alternatively, you can use an immersion blender.
In the bowl of a stand mixer, dissolve the 1 teaspoon granulated sugar in ¼ cup warm water and sprinkle over the 4½ teaspoons active dry yeast. Let stand until yeast begins to foam, about 5 minutes. Add the beet mixture, 1 egg, ½ cup granulated sugar, ¼ cup butter, and ¼ teaspoons fine grain sea salt, and mix together with paddle attachment.
Add about half of the 4 1/2 cups all purpose flour along with the 2 tablespoons cocoa powder and continue to mix with the paddle attachment. Switch to the dough hook attachment and add the rest of the flour. Knead until dough comes together and forms a ball, about 5 minutes. Transfer dough to a lightly greased bowl and cover in a warm room. Let rise for 1½ to 2 hours until doubled in size.
In a small bowl, use a fork to mix together the ¾ cup sugar, 1 tablespoon cinnamon, and 1 teaspoon cocoa powder for the filling.
Divide dough into two portions. On a lightly floured surface, roll each portion into a rectangle about 13" by 20". Spread half of the softened 6 tablespoons unsalted butter on each rectangle and sprinkle with cinnamon sugar mixture, leaving a ½-inch border around the edges. Roll up dough lengthwise into tight logs and place the logs seam side down. Gently cut each log into 12 pieces using a serrated edge knife. Place all 24 rolls on a large parchment lined baking sheet.
Lightly cover rolls and allow rolls to rise again for 30 minutes.
Preheat oven to 400°F. Bake for 14-18 minutes, until light golden. Do not over bake.
While rolls are cooking, whisk together 2 cups confectioners sugar, 1-2 tablespoons milk, and reserved beet juice in small bowl. Add just enough milk to get it to your preferred consistency.
When rolls, are done cooking, remove from oven and allow to cool slightly before covering with icing.
Store leftovers in the freezer wrapped in foil and reheat in the oven when ready to serve.
Notes
If your home is cold, you can create a warmer environment for your dough to rise in the oven. Just preheat the oven to 150-200 degrees F. As soon as it hits temperature, turn the oven off and crack the door. Let the dough rise in there and once the inside of the oven starts cooling off to room temperature, close the door and allow it to finish rising.
The amount of flour you will use will depend on your dough. You'll want it to be elastic and not overly sticky.
You can roast the beets in the oven as described above, your you can cook in Instant Pot on rack with 1 cup of water for 15 minutes or until very soft.