To make the dough: Place the yeast and ½ teaspoon sugar in a small bowl. Add the lukewarm water and stir gently to mix. Set aside until foamy, 5 to 10 minutes.
In a stand mixer fitted with a dough hook, mix together the flour, ⅓ cup sugar, and 2 teaspoons vanilla.
In a medium saucepan, scald the milk (bring almost to a boil, until milk is just simmering). Allow to sit for 1 minute to cool just slightly. With mixer on low, add the water-yeast mixture, milk, and melted butter. Add eggs one at a time.
When the dough begins to come together, after 2 to 3 minutes, turn off mixer and scrape down the sides. Raise the speed to high and mix for another 5 to 10 minutes until the dough is shiny and elastic. Place dough in a greased bowl with a damp towel on top. Allow to rise until it has doubled, about 1 to 2 hours.
While dough is rising, make the filling and the topping. To make the filling: Beat the softened cream cheese, powdered sugar, 1 teaspoon vanilla, baking powder, and salt. Fold in the sprinkles.
To make the crumb topping: Combine the sugars in a small bowl. Stir in the melted butter and the flour. Add the sprinkles and combine using fingers. Prepare three 8 ½-by-4 ½-inch greased loaf pans. Note: you can also make two larger round babkas that can be baked on baking sheets.
Cut the dough into three equal parts (use a food scale for precision). Roll out one part into a rectangle. Spread with one-third of the cream cheese filling and roll up along the shorter side (to create more swirls inside).
Once the dough is formed into a swirled log, cut it straight down the middle so the filling is exposed. Cut ½ inch off each end. Layer each cut piece on top of one another and twist. Place in a greased loaf pan. Repeat with other two pieces of babka dough.
Lightly drape a kitchen towel over the top of pans. Allow to rise another 30 minutes.
Preheat oven to 350°F while the dough rises. Before baking, add one-third of the crumb topping on each babka. Bake for 30 to 35 minutes. The edges should be slightly brown and the middle should be slightly doughy.
Allow to cool for 5 to 10 minutes. Using a butter knife, loosen sides of the babkas from the pans and place on top of parchment wire rack to cool. Place babkas on a platter. To make the frosting: Combine confectioners’ sugar, milk, and salt in a bowl. Drizzle on top of each babka.
Note: It is best to use jimmies or oblong sprinkles, not round (nonpareil) sprinkles.
Short cut: If you don’t want to make the crumb topping and frosting, you can skip those steps. After the babka has baked and cooled, combine 1 cup (approximately 6 ounces) white chocolate chips with 1 tablespoon vegetable oil in a microwave-safe bowl. Microwave for 30-second intervals, mixing in between with a small spatula, until completely melted and smooth. Drizzle melted white chocolate on top of each babka and allow to set.