Place the yeast, ½ teaspoon of sugar, and ⅓ cup of warm water in a small bowl. Stir, and let sit for 10 minutes to proof yeast.
Add the flour, 3 tablespoons of sugar, and salt to the bowl of a stand mixer, and whisk to combine. Add the egg, 6 ½ tablespoons of melted butter, and the yeast/water mixture to the flour mixture. With the dough hook attached to the mixer, mix on low for 10 minutes.
Remove the dough and add to a bowl that has been greased with oil. Top with a towel and set aside for 1 ½ to 2 hours to rise.
Make the sugar syrup by adding the remaining 4 tablespoons of sugar and the remaining ¼ cup of water in a small saucepan, bringing it to a boil over medium heat, and whisking until the sugar dissolves. Remove from heat and set aside.
Make the cinnamon filling by mixing the remaining ⅓ cup of butter, softened, the brown sugar, and cinnamon in a small bowl, until a grainy, spreadable paste is formed.
Grease a 9x5-inch loaf pan.
Turn the dough out onto a floured work surface, and roll out to a roughly 8x12-inch rectangle.
Using a spatula, spread the cinnamon filling over the dough, covering it entirely.
With a long end nearest you, roll the dough into a log, ending at the opposite long side. Press to seal the ends and rest the dough on its seam.
Using a serrated knife, cut the dough in half lengthwise, exposing the cinnamon filling.
Place the 2 halves with the filling side facing up lengthwise in front of you. Press the 2 ends together of the 2 halves together. Lift the right piece over the left, and then lift the left piece over the right and continue forming a simple plait. Press the ends together when finished.
Gently lift the bread into the prepared pan. Cover with a cloth and allow to rise for 1 hour more.
Preheat the oven to 350 F.
Bake the babka for 35 to 40 minutes, until the edges are slightly brown and the top is golden.
Brush the top of the babka with the sugar syrup. Allow it to cool for 10 minutes, then remove from pan.