Make the sweet starter (the night before)

In a small jar, stir together the 20 g starter, 60 g bread flour, 30 g water, and 15 g sugar until smooth. Cover loosely and leave at room temperature overnight, 8 to 12 hours, until very bubbly and roughly 2.5 to 3 times its size. It should look domed and smell mild and sweet.

Make the tangzhong (next morning)

Whisk the 30 g bread flour into the 150 g milk in a small saucepan until smooth. Cook over medium-low heat, stirring constantly, until it thickens into a soft, pudding-like paste, about 2 to 3 minutes. Scrape into a bowl and let it cool to just warm.

Mix the dough

In a large bowl or stand mixer, combine all the sweet levain, the cooled tangzhong, 125 g pumpkin puree, the egg, 30 g milk, and 50 g sugar. Add the 400 g bread flour and 9 g salt. Mix to a shaggy dough, then knead until smooth and stretchy, about 8 to 10 minutes by machine (longer by hand). Add the reserved flour only a little at a time, and only if the dough is too loose. It should be soft, supple, and slightly tacky, not stiff.

Add the 70 g soft butter a little at a time, letting each addition work in before the next. Keep kneading until the dough is glossy, smooth, and pulls away from the bowl, about 5 more minutes. It will stay soft and a touch tacky. Resist adding more flour.

Bulk ferment

Cover and let rise at warm room temperature, ideally 76 to 80°F, until puffy and increased about 30 to 50% in volume, roughly 3 to 5 hours. Enriched sourdough moves slowly, so watch the dough, not the clock.

Chill

Cover and refrigerate 1 to 3 hours. Cold dough rolls out cleanly and won't fight you.

Make the filling and shape

Stir the 85 g soft butter, 135 g brown sugar, 25 g pumpkin, 8 g cinnamon, 5 g pumpkin pie spice, 8 g flour (or 5 g cornstarch), and a pinch of salt into a thick, spreadable paste. Keep it soft, never melted.

Turn the chilled dough onto a lightly floured surface and roll into a rectangle about 18 by 15 inches. Spread the filling evenly to the edges. Starting from a long side, roll into a tight log. Cut 12 even rolls with unflavored dental floss: slide it under the log, cross the ends over the top, and pull. Arrange in a greased or parchment-lined 9x13 pan.

Proof (choose one)

Overnight (recommended): cover and proof at room temperature until the rolls look about three-quarters puffed, then refrigerate overnight. Bake straight from the fridge in the morning, adding a few minutes to the bake time.

Same day: cover and proof at warm room temperature until very puffy and touching, and a gentle fingertip press leaves a slow-moving dent, about 2 to 4 hours.

Bake

Heat the oven to 350°F. If using, slowly pour the 80 g warm half-and-half or cream over and between the proofed rolls for an extra-gooey finish. Bake 25 to 30 minutes, until golden and about 190 to 195°F in the center. Do not bake until deeply browned, or they'll dry out.

Make the frosting

Brown the 56 g butter in a small pan over medium heat, swirling, until it smells nutty and turns golden with brown flecks. Cool until just warm. Beat with the 113 g cream cheese until smooth, then beat in the 120 g powdered sugar, 30 g maple syrup, vanilla, optional maple extract, and a pinch of salt until creamy.

Finish

Let the rolls cool 10 to 15 minutes, then spread the frosting so it melts slightly into the warm tops. Serve warm.