Warm the milk to about 95–105°F (warm to the touch, not hot). Reserve 1 tablespoon of the warm milk for brushing later, then combine the remaining 1 + cup warm milk with 4 ½ teaspoons active dry yeast in a small bowl. Stir gently and let sit 5–10 minutes, until foamy and aromatic.
In the bowl of a stand mixer fitted with the dough hook, add 3 ¾ cups unbleached all-purpose flour, ⅓ cup sugar, and 2 teaspoons salt. Mix briefly on low to combine the dry ingredients.
Pour the milk/yeast mixture into the mixer bowl with the dry ingredients. With the mixer on the lowest speed, mix 1–2 minutes until a soft, moist dough forms on the hook and most of the flour is incorporated.
If the dough seems too dry or there are dry pockets of flour, add additional warm milk 1 tablespoon at a time until all the flour is moistened and the dough comes together on the hook.
Increase the mixer to a low-medium speed (the highest speed your machine recommends for dough; for many machines this is speed 2). Knead until the dough is smooth, elastic, and slightly tacky but not sticky—about 6–8 minutes total kneading time. The dough should pull away from the sides of the bowl and form a smooth ball on the hook.
Remove the dough from the mixer and place it in a large clean plastic bag or an airtight container, leaving room for expansion. Let the dough sit at room temperature for 30 minutes.
After 30 minutes, refrigerate the dough for 8 hours or overnight to develop flavor and make the dough easier to handle.
When ready to shape, lightly flour a work surface. Remove the chilled dough from the refrigerator and dust the top lightly with flour.
Cut the dough in half horizontally so you have two pieces approximately equal in size. Wrap one half and return it to the refrigerator while you work with the first half.
On the floured surface, roll the first half into a rectangle about 24–26 inches long and 15–18 inches wide. Keep the surface and the dough well-floured to prevent sticking. If the dough resists, let it rest 5–10 minutes and continue; do not overwork it.
Fold the top half of the rectangle down toward the bottom so you have a single folded sheet about half the original length. Turn the folded dough so the long side faces you.
Using a ruler or bench scraper, cut the folded dough into strips 3 to 4 inches wide along the short dimension, then cut each strip in half crosswise to make manageable pieces (about 6–8-inch lengths). Repeat with the second half of the dough after you have shaped the first batch.
Shape each strip into a croissant-style roll: take one strip, stretch it slightly if needed, then roll it from one end toward the other into a tight coil or crescent, finishing with the seam side down on the prepared baking sheet. Place shaped rolls on parchment-lined baking sheets, leaving room for expansion.
Lightly brush the tops of the shaped rolls with the 1 tablespoon of reserved warm milk (no egg wash required). Cover the baking sheets loosely with plastic wrap or a clean kitchen towel and let the rolls rise, uncovered at room temperature, for 3–4 hours, until noticeably puffy and nearly doubled (or up to tripled) in size. A warm, turned-off oven with a pan of hot water can help create a humid rise environment.
About 15 minutes before baking, preheat the oven to 375°F (190°C). When the rolls have risen, brush them once more lightly with the reserved milk if any remains, or with a little additional milk taken from your supply if desired.
Bake the rolls on the middle oven rack at 375°F for 18–22 minutes, rotating the pans front-to-back halfway through, until the rolls are deeply golden and sound hollow when tapped. Baking time will vary with oven and roll size—watch for an even, deep color.
Remove the rolls from the oven and transfer to cooling racks. Cool at least 15 minutes before serving.
Storage: the rolls are best the day they are made. To freeze, cool completely, then place in an airtight container or freezer bag. Thaw overnight in the refrigerator or at room temperature, and reheat in a 350°F oven for about 8 minutes before serving.