Mix 2 ½ cups of the flour and yeast in a stand mixer, 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 120°F to 130°F
With mixer on low, pour in liquids and mix until incorporated. Add eggs one at a time until fully incorporated
Clean off paddle and switch to a dough hook. Mix in the remaining 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 until each piece of butter is combined entirely
Knead for eight minutes on low, stopping every couple of minutes to scrape the dough off the hook. The dough should completely clear the sides of the bowl. If it is too sticky, add additional flour 1 tablespoon at a time as needed. If the dough seems too dry, sprinkle with a little water
On a lightly floured surface, knead the dough a few times by hand until smooth and elastic. Form the dough into a ball and place into a bowl sprayed lightly with cooking spray. Give the top of the dough a little spray and then cover the bowl with plastic wrap and refrigerate for 1 hour
While the dough is chilling, make the filling. In a medium bowl, combine the brown sugar, cinnamon, and butter until crumbly
Roll the dough into a 10 x 20 inch rectangle. Spread the filling on half of the dough. Fold the dough in half covering the filling and press the dough down firmly so the dough will stick together. Trim any odd edges and then cut the dough into three long strips. Press the tops of the strips together and then braid. Press the ends together at the bottom. Gently stretch the braid until it’s back to about 20 inches
Shape it into a circle/oval and press the edges together then transfer it to a parchment lined baking sheet. Cover and let it rise at room temperature for about 1 hour
While the dough is rising, preheat the oven to 350°. Bake the cake until it is golden brown and sounds hollow when tapped, about 20 to 35 minutes
Remove from oven and let cool for 15 minutes while you make the icing
If you want to hide the baby (or bean)in the cake, make a small slit in the bottom of the cake and put the miniature plastic baby in after the cake has cooled.
To make the icing, Icing mix the powdered sugar, milk, and vanilla until smooth (add a little milk if too thick or sugar if too thin).
Spread the icing over top of the still warm cake and Immediately sprinkle on colored sugar
Notes
Do not use Active Dry yeast - the proofing times will be wrong.