Step 1: Prepare the Dough

1. Activate the yeast: In a small bowl, combine the warm milk (about 110°F) and sugar. Sprinkle the active dry yeast on top and let it sit for about 5 minutes until it becomes frothy. If the mixture doesn’t bubble, it might mean the yeast is old, so try again with fresh yeast.
2. Mix the dough: In a large bowl, combine the flour and salt. Make a well in the center and pour in the activated yeast mixture, followed by the softened butter, eggs, and vanilla extract. Using a spoon or your hands, mix the dough until it comes together.
3. Knead the dough: Transfer the dough to a floured surface and knead for about 8-10 minutes, until the dough is smooth and elastic. If it’s too sticky, add a little extra flour, but be careful not to add too much. You want the dough to be soft and slightly tacky.
4. Let the dough rise: Place the dough in a greased bowl and cover it with a clean kitchen towel. Let it rise for 1-1.5 hours, or until it doubles in size. This is when the dough gets its airy texture.

Step 2: Prepare the Filling

1. Mix the sugar and cinnamon: In a small bowl, combine the sugar and cinnamon. This will be sprinkled over the dough to create that sweet, donut-like flavor.
2. Get the jam ready: If your jam is a little chunky, you can heat it gently in the microwave for a few seconds to loosen it up. You want it spreadable but not too runny.

Step 3: Shape the Babka

1. Roll out the dough: Once the dough has risen, punch it down to release the air. Roll it out on a floured surface into a large rectangle about 12×18 inches. You want the dough to be nice and even.
2. Spread the jam: Spread the jam evenly over the dough, leaving about 1 inch around the edges free. Sprinkle the cinnamon-sugar mixture over the jam.
3. Roll the dough: Starting from one long side, carefully roll the dough into a log, as if you’re making a cinnamon roll. Make sure the seam is on the bottom.
4. Cut and braid the dough: Using a sharp knife, slice the rolled dough in half lengthwise to expose the swirl. Place the two pieces on top of each other, cut-side facing up. Twist them together into a braid, tucking the ends underneath to seal.
5. Transfer to a baking pan: Place the braided dough into a greased loaf pan. Cover with a towel and let it rise again for 30-45 minutes.

Step 4: Bake the Babka

1. Preheat the oven: Preheat your oven to 350°F (175°C).
2. Egg wash: Brush the top of the babka with the beaten egg wash to give it a beautiful golden finish.
3. Bake: Place the babka in the oven and bake for 30-35 minutes, or until the bread is golden brown and sounds hollow when tapped on top.
4. Cool: Let the babka cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.

Step 5: Finish with Powdered Sugar

Once the babka has cooled, dust the top with powdered sugar for a sweet finishing touch.