Mix 2 ½ cups flour and yeast in mixing bowl, using the paddle attachment, on low for about 30 seconds.
Heat milk, sugar, and salt in a small saucepan over medium heat until sugar is dissolved and milk is between 120º to 130º.
With mixer on low, pour in liquids and mix until incorporated. Add eggs one at a time. Continue mixing until a shaggy dough forms.
Clean off paddle and switch to dough hook. Mix in the remaining 1 cup flour a little at a time, adding more or less flour as needed to make a soft dough.
Add the softened butter, a piece at a time, kneading until each piece of butter is absorbed. Knead for eight minutes on low. The dough should completely clear the sides of the bowl. If it is too sticky, add additional flour 1 tablespoon at a time, mixing in thoroughly before determining if more flour is needed. If the dough seems too dry, spritz with water from a spray bottle a couple of times, mixing in thoroughly before determining if more water is needed. Every 2 minutes, stop the machine, scrape the dough off the hook, and then continue kneading.
Turn the dough out onto a lightly floured surface and knead a few times by hand to be sure it’s smooth and elastic. Form the dough into a ball. Place dough into a greased bowl. Turn once so greased surface is on top. Cover bowl with plastic wrap and refrigerate for 1 hour.
While the dough is chilling, make cinnamon filling. Combine the brown sugar and ground cinnamon. Combine butter with cinnamon mixture and mix well.
Roll the chilled dough into a 10 x 20 inch rectangle. Spread the filling on half of the long side of the dough. Fold the dough in half covering the filling. Pat dough down firmly so the dough will stick together.
Cut dough into three long strips lengthwise. Press the tops of the strips together and braid the strips. Press the ends together at the bottom.
Gently stretch the braid so that it measures 20 inches again. Shape it into a circle/oval and press the edges together. If desired, hide the miniature plastic baby in between the ends of the circle before sealing together in a ring. (You can also make a slit in the bottom of the cake and put the baby in after the cake is baked.)
Transfer the ring to a parchment lined or greased baking sheet. Cover the dough with plastic wrap and let it rise until doubled, about 1 hour.
While the dough is rising, preheat the oven to 350º. Bake the cake until it is golden brown, 20–35 minutes. (It should feel firm to the touch, and have an internal temperature of about 190°.) Remove from oven and let cool 10 minutes on baking sheet and then place it on a cooling rack to cool completely before icing.
Icing: In a small bowl, mix powdered sugar, milk and vanilla until smooth (add additional milk if mixture is too thick or powdered sugar if too thin). Spoon icing over top of the cake. Immediately sprinkle on colored sugar, alternating between the three colors.