Gently heat the milk to a warm body temperature (about 100 degrees). In a small bowl, whisk the warm milk with the yeast and sugar. Set aside for 5 to 10 minutes until bubbly.
In the bowl of a stand mixer fitted with the paddle attachment, add the creme fraiche, eggs and vanilla extract. Mix on low to combine. Add the yeast mixture, melted butter and 1 teaspoon of kosher salt.
On low speed, add the flour one cup at a time, and mix until the dough comes together. It’s okay if it’s not completely smooth, there is more kneading to come!
Remove the dough from the bowl, and place on a large counter or work surface. Divide the dough into 3 equal portions. Working with one portion at a time, add desired amount food coloring and knead by hand to distribute evenly. This will take some time, and it’s okay if the color is streaky — it’s more important that the dough is smooth. Well-fitted disposable gloves are great to prevent staining your hands.
Place the colored dough balls into three small, greased bowls and cover with plastic wrap. Let them rise for 45 minutes to an hour in a warm spot. The dough is ready when it is almost doubled in size.
In a small bowl, stir together the brown sugar, ground cinnamon and 1 teaspoon kosher salt.
Lightly flour the work surface and roll each dough portion out to roughly 18 by 6-inch rectangles. Evenly scatter the brown sugar filling over each rectangle. Roll each piece of dough into a spiraled log, like a cinnamon bun, moistening the end with water to seal.
Once all three pieces are rolled, joint the ends of all three together with some water and braid the logs together. Move the braid to a parchment lined baking sheet and form the braid into a large ring, joining the ends together by firmly pressing them together with warm water.
Cover the sheet with plastic wrap and let rise for 20 to 30 minutes. Preheat your oven to 375 degrees F.
Once the ring has puffed slightly, bake at 375 degrees F for 30 to 35 minutes. It should be golden brown on top, and not gooey when poked with a knife. Let cool.
FOR THE ICING:
In the bowl of an electric mixer fitted with a paddle attachment, beat together the cream cheese, powdered sugar, vanilla extract and salt until smooth. Add the warm water. Divide the icing into three portions and tint with the desired amount of food coloring.
Ice the cooled cake as desired. To get a stringy effect, use 3 piping bags with very small tips and zig zag the icing over the cake alternating the colors.