About 4 hours before mixing the dough, feed your sourdough starter or build a levain at a 1:1:1 ratio with the starter, flour, and water. Let it stand at 77 to 80 degrees F until it looks doubled, bubbly, and airy throughout.

Add the active sourdough starter, warmed milk, bread flour, salt, sugar, egg, butter, and vanilla to the bowl of a stand mixer and knead on low speed for 8 to 10 minutes. Keep mixing until the dough stretches thin enough to pass the windowpane test and feels smooth instead of shaggy.

Move the dough to a covered container and set it in a warm place for bulk fermentation. At about 80 degrees F, it should roughly double in 4 to 5 hours and look fuller, softer, and lightly domed.

Stir together the softened butter, flour, brown sugar, cinnamon, and nutmeg in a small bowl, then peel, chop, and thoroughly dry the peaches with paper towels. The filling should look spreadable, and the fruit should feel dry on the surface rather than glossy or wet.

Press the risen dough down gently and roll it into a 13.5 by 10 inch rectangle. Keep the thickness even so the stacked sections bake at the same rate.

Spread the spiced filling over the dough and scatter the dry peaches evenly across the top, blotting away any extra juice if needed. You want an even layer of fruit without wet patches that can make the strips slide.

Slice the dough into three 4.5 inch strips, stack the strips with the filling side facing up, and cut that stack crosswise into five 2 inch sections. Work neatly so the layers stay aligned and the filling stays inside the stack.

Turn each section onto its side and arrange the stacks upright in a parchment-lined 9 by 5 inch loaf pan with the fruit layers facing the long side of the pan. The pieces should sit like shingles, snug but not crushed, so the loaf can puff between the layers.

Cover the pan and let the loaf rise at about 80 degrees F for about 1 hour. It should look visibly puffy and pillowy before it goes into the oven.

Cut the cold butter into the flour, brown sugar, and cinnamon until coarse crumbs form, then sprinkle the streusel over the top. The crumbs should stay a little chunky so the topping bakes into a textured crust.

Bake the loaf at 350 degrees F for 75 minutes, leaving it uncovered for the first 45 minutes and loosely tenting it with foil for the final 30 minutes. The top should be deep golden, the kitchen should smell like warm cinnamon peaches, and the center should look set rather than damp.

Whisk the powdered sugar, milk, and vanilla extract together, then drizzle the icing over the slightly cooled loaf before serving. The glaze should run over the ridges and melt lightly into the warm layers.