In the bowl of a stand mixer with a dough hook, combine the whole wheat flour, all purpose flour, about 3/4 of water, active sourdough starter, honey, and salt.
Mix the dough on low speed until incorporated, about 2 minutes.
Switch to the dough hook and mix on medium speed for 4-5 minutes, or until the dough starts to strengthen and clump around the dough hook. The dough should be strong, but it won’t completely leave the bottom of the bowl as it mixes, and will still look a little shaggy. But the majority of the dough should be around the dough hook. If the dough is not medium soft, add a little bit of the extra water at a time and mix until it is.
Cover the dough and allow it to rest for 10 minutes.
Turn the dough on low, and slowly add the butter, 1 piece at a time, allowing the butter to be fully incorporated into the dough before adding the next piece of butter. It will take about 3-5 minutes. It will be a soft but strong dough.
Bulk Fermentation
Place the dough in a bowl and cover with a cloth. Set it in a warm place (about 75 F) and let it sit for 30 minutes.
Do the first set of stretch and folds.
Recover the dough, and allow it to sit for an additional 30 minutes.
Do the second set of stretch and folds.
Recover the dough, and allow it to sit for 30 minutes.
Do the third set of stretch and folds.
Recover the dough and allow it to sit for an hour and a half.
Shaping
Place the bowl of dough into the fridge for an hour and a half to chill the dough. You can technically skip this step, but it really helps the dough to be easier to handle when shaping.
Line a baking sheet with parchment paper or a silpat mat and set aside.
At this point, the dough should be well risen, bubbly, light but also firm.
Lightly flour your work surface and gently dump out the dough.
Divide the dough into 9 pieces, each about 100g. Cover with plastic wrap and allow to rest for 20 minutes to help the gluten relax, making it easier to shape into buns.
Shape each dough portion into a log, about 6” inches long (or into a smooth ball for hamburger buns!).
Begin by stretching and patting the dough into a rough rectangle (with the long side horizontal) . Fold the ends in. Take the top edge and roll towards your body, pressing the dough into itself and away from you (like how you would shape a pretzel). Continue rolling and pressing until you form a tube. When you have reached the bottom, seal the seam.
Place the buns on the prepared baking sheet with the seam side down, about 1 to 2 inches apart so the buns will barely touch after they have risen.
Final Proofing
Cover with plastic wrap and a tea towel and allow to rest at room temperature for 1-3 hours, or until doubled. (The timing will depend on the activity level of your starter and the temperature of your house, so watch the dough here, not the clock)
Baking
Preheat the oven to 400F (you will also need steam for baking, so prepare for this now).
Brush a little milk over the buns, as this will help them develop an even golden color when baking.
Remove the steam and bake for an additional 25-35 minutes, turning the pan halfway through baking, until the buns are nicely browned.
Remove from the oven and brush with melted butter. This helps them maintain their soft texture.
Cool completely on a wire rack before separating the buns.