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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess.
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To make the dough: In the bowl of a stand mixer fitted with the dough hook, combine all the dough ingredients and mix on medium speed for about 1 minute, pausing to scrape the sides of the bowl halfway through.
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Increase the speed to medium-high and knead until a cohesive, tacky dough forms, about 3 more minutes, scraping the bowl halfway through.
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Transfer the dough to a lightly greased bowl, cover, and let rest until slightly puffy, 60 to 75 minutes; the dough will not fully double in size.
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Line an 8" square pan with a parchment paper sling and lightly grease the parchment.
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To shape the whole wheat rolls: Gently deflate the dough and transfer it to a lightly floured surface. Using a bowl scraper or bench knife, divide the dough into 16 pieces (about 49g each).
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Shape each piece into a rough ball by stretching the edges outward and bringing them back toward the center, gathering them into a dumpling-like shape. Place the pieces seam-side down and roll each under a cupped hand to form a smooth ball. Use additional flour as needed to prevent sticking. If the dough is sliding around on the work surface, lightly dampen the surface with water to create more traction and help you roll the dough snugly. See our video, How to shape rolls, for a visual demonstration of this technique.
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Place the shaped rolls into the prepared pan in a 4 x 4 pattern, cover, and let rise until slightly puffy, about 1 hour; when pressed with a floured finger, an indent should remain. Toward the end of the rising time, preheat the oven to 350°F with a rack in the center.
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To top and bake the rolls: When the rolls have risen, brush the tops gently with the egg wash, then bake for 28 to 32 minutes, until well browned on top and pale around the edges of the pan. A digital thermometer inserted into the center of one of the middle rolls should read at least 205°F when they’re done. While the rolls are baking, prepare the salted honey butter.
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To make the salted honey butter: In a small, microwave-safe bowl, heat all the ingredients in 15-second increments at 50% power, stirring between each, until melted. Set aside.
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Remove the whole wheat rolls from the oven and let them cool in the pan for 10 minutes before using the parchment sling to transfer them onto a wire rack.
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While the rolls are still warm, brush the tops and sides with salted honey butter; serve warm or at room temperature.
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Storage information: Store cooled whole wheat rolls, tightly wrapped, at room temperature for several days; freeze for longer storage.
