Proof the yeast by combining warm milk, a teaspoon of sugar, and yeast in the mixer bowl. Let sit until foamy, about 10 minutes.
Add the flour, remaining sugar, and salt to the yeast mixture. Mix with the dough hook on low, then add eggs, yolk, and vanilla. Knead until combined.
With the mixer running, add the cubed butter one piece at a time, waiting for each to incorporate. Knead until the dough is smooth and pulls away from the bowl.
Place the dough in a greased bowl, cover, and let rise at room temperature until doubled in size, about 1.5 to 2 hours. (Or refrigerate overnight for better flavor).
While the dough rises, mix softened butter, brown sugar, powdered sugar, and cinnamon together to form a spreadable paste for the filling.
If using the streusel, combine flour, sugar, and cinnamon in a bowl, then cut in cold butter until crumbly. Refrigerate until ready to use.
Punch down the risen dough and roll it into a 12x18-inch rectangle. Spread the cinnamon filling evenly over the dough, leaving a 1-inch border on one long edge.
Roll the dough tightly into a log starting from the opposite long edge. Chill in the freezer for 15-20 minutes if it becomes too soft, then cut the log in half lengthwise.
Place the two halves with the cut sides facing up, cross them over each other in a twist, and pinch the ends to seal. Transfer the twist to a parchment-lined 9x5-inch loaf pan.
Cover the pan and let the babka rise at room temperature until puffy and domed, about 1 to 1.5 hours. Meanwhile, preheat the oven to 350°F (175°C).
Scatter the streusel over the risen loaf, if using. Bake for 40 to 45 minutes until deep golden brown and the internal temperature reaches 190°F (88°C).
While the babka bakes or immediately after, boil the sugar and water together to make a simple syrup. Brush the hot loaf generously with the warm syrup as soon as it comes out of the oven.
Let the babka cool in the pan for 20 minutes, then transfer to a wire rack to cool completely before slicing and serving warm or at room temperature.