Measure out the milk and heat until lukewarm (I like to do this in a glass measuring cup in the microwave). Sprinkle the yeast on top of the warm milk and let it sit for five minutes, until it is foamy.
Combine the butter and sugar in the bowl of a stand mixer and beat with a paddle attachment until light and fluffy. Add the eggs and maple syrup and mix until well-integrated.
Measure out the flour and whisk in the salt.
With the mixer running on low, alternate between adding the flour and the milk, until both are incorporated.
Stop the mixer and switch to the dough hook. Knead for 4 to 6 minutes on low, until the dough comes together and forms a soft, smooth ball.
Butter a large bowl and set the dough in it. Cover with plastic wrap and set the bowl in a warm spot to rise for about two hours, or until the dough has doubled (remember that if your ingredients were quite chilly, the rising time might be even longer).
Once the dough has doubled, gently deflate it. At this point, you can either refrigerate your dough and pick up the following day, or you can proceed.
Divide the dough into two equal sized portions.
To make the babka, roll out the dough into a 18 x 12 inch rectangle. Brush with melted butter, apply a thin layer of apricot jam, and evenly scatter half the walnuts (trying to leave about an inch margin on one short edge).
Starting with the other short edge, carefully roll the dough into a tight log, working towards the bare margin.
Once the dough is rolled, slice it in half down the middle, leaving top inch of the roll uncut and still attached.
To twist the babka into its traditional shape, grab the ends and twist them opposite one another towards the outsides of the loaf.
Nestle the twisted loaf into a loaf pan, cover it with plastic and let it rise. While it rises, preheat the oven to 350 degrees F. Brush the babka with the egg wash and bake for 35 to 40 minutes, until an instant read thermometer says that the loaf is approximately 200 degrees F inside.
To make the sticky buns, roll out the dough into a 12 x 10 rectangle. Brush with melted butter, sprinkle evenly with the sugar and cinnamon, and then spread out the remaining walnuts.
Starting with the long side, roll the dough. Cut it into buns about 1 1/2 inches thick and arrange them in a pie plate.
Let the buns rise and then par-bake at 350 degrees F for 10-12 minutes.
Remove pan from oven and let cool. Tuck it into a large ziptop bag and freeze.
The night before you want to eat the sticky buns, move the pan from the freezer to the fridge to defrost.
In the morning, heat the oven to 350 degrees F. Bake the buns for 15 or so minutes, until they're quite golden. While they bake, melt the butter and combine the powdered sugar, milk, cinnamon, and vanilla.
Brush with melted butter while the buns are still hot. Once they've cooled just a bit, drizzle on the glaze.
Serve warm or room temperature.