In a stand mixer bowl, combine all dough ingredients except butter. Mix on low speed until fully combined and no dry flour remains.
Gradually add the butter, one tablespoon at a time, mixing each addition in before adding the next.
Once all butter is incorporated, increase mixer speed slightly and mix for about 20 minutes, until the dough no longer sticks to the bowl and passes the windowpane test.
Scrape down the sides of the bowl. Cover with plastic wrap or a damp tea towel and let the dough rest for 30 minutes.
Perform 2–3 sets of stretch and folds, resting 30 minutes between each set.
Let the dough ferment at room temperature (~72°F/22°C) until it increases by 50% in volume (usually 6–8 hours). The dough should be puffy but not doubled.
Transfer the dough to the refrigerator and chill overnight to improve flavor and make shaping easier.
Meanwhile, prepare the filling: In a small bowl, mix softened butter, brown sugar, cinnamon, and flour until it forms a thick paste. Set aside.
Remove the chilled dough from the fridge. Lightly flour your work surface and the top of the dough. Roll into a 14 x 18-inch (35 x 45 cm) rectangle.
Spread the filling evenly over the dough, leaving a small border around the edges.
Roll the dough tightly from the long side, like a cinnamon roll. Slice the roll lengthwise down the center to create two long strips with the filling exposed.
Pinch the top ends together, then twist the two strands around each other to form a braided loaf. Place carefully into a parchment-lined 9 x 5-inch loaf pan (or 8.5 x 4.5-inch for a taller loaf).
Cover the loaf and let it rise at room temperature for 2–4 hours, until slightly puffy (volume should increase but not double).
(Optional) Refrigerate the shaped babka overnight for enhanced flavor. If refrigerated, allow it to come to room temperature for about 1 hour before baking.
Preheat the oven to 350°F (175°C). Bake for 45–55 minutes, checking at 45 minutes. Babka is done when the center reaches 200°F (93°C). If it browns too quickly, cover loosely with foil.
Remove from the oven and cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.