Feed your sourdough starter 4–6 hours before mixing, using a 1:1:1 ratio of starter, flour, and water. Place in a warm spot (75–80°F / 24–27°C) until doubled, bubbly, and domed on top.
Whisk together warm milk, active starter, and sugar until the starter is dissolved and slightly frothy, then whisk in melted butter. Add bread flour and salt, mixing until all flour is hydrated and no dry patches remain — the dough will look shaggy.
Cover the bowl with a damp towel or plastic wrap and rest the dough at room temperature for 30–60 minutes. The dough will become smoother and more cohesive, making kneading much easier.
Knead on a lightly floured surface for 8–10 minutes by hand (or 5–7 minutes in a stand mixer on medium) until smooth, elastic, and passing the windowpane test. Shape into a ball, place in a lightly greased bowl, and turn to coat.
Cover and let the dough rise in a warm, draft-free spot (75–78°F / 24–26°C) for 6–8 hours until doubled in size, puffy, and slowly springing back when gently pressed. Timing varies with kitchen temperature.
Combine softened butter, brown sugar, cinnamon, and a pinch of salt in a small bowl, mixing with a fork until a smooth, spreadable paste forms. Set aside at room temperature.
Gently deflate the risen dough, then roll it out on a lightly floured surface into a 9×12–14 inch rectangle; spread the cinnamon filling evenly, leaving a 1/2-inch border along one long edge, then tightly roll into a log and pinch the seam and ends closed.
Place the log seam-side down in a greased 9×5-inch loaf pan, cover loosely, and proof at room temperature for 2–4 hours until the dough rises about 1 inch above the rim and slowly springs back when pressed.
Preheat the oven to 350°F (175°C) with a rack in the center about 30 minutes before baking. Gently brush the top of the proofed loaf with beaten egg wash and optionally sprinkle with coarse sugar.
Bake for 45–50 minutes, rotating halfway through, until deep golden brown and an instant-read thermometer reads 190–200°F (88–93°C) in the center; tent loosely with foil if the top browns too quickly.
Cool in the pan for 10 minutes, then transfer to a wire rack and cool for at least 1 hour before slicing with a serrated knife — cutting too soon results in a gummy crumb.