In a saucepan, add the milk, butter, sugar, and salt. Set over medium heat and bring to about 90ºF (32ºC). The milk should be warm, and the butter just starting to melt.
Transfer the mixture to a large bowl or stand mixer and add the sourdough starter. Using a whisk or the whisk attachment on a stand mixer, stir to combine.
Add about half of the flour (2 cups/250 grams) Mix on low-medium speed until combined.
Add the egg and mix until incorporated.
Switch your whisk attachment for your dough hook attachment and slowly add the remaining flour to the dough. Continue to mix until a shaggy dough forms; the dough should come together and pull away from the sides of the bowl*.
Knead by hand (8-10 minutes) or using the dough attachment on your mixer (5-7 minutes). The dough should be soft, slightly tacky, and elastic.
Place the dough in a greased bowl. Cover tightly with plastic wrap and let rise at room temperature for 8-12 hours, until puffy and nearly doubled in size (it won’t quite double).
Turn the dough onto a lightly floured surface and divide into 8-10 equal pieces (for equal-sized buns, divide the weight of the proofed dough by 8, and weigh each piece).
Shape each into a log by flattening slightly into a rectangle, then roll each piece tightly into a 5-inch log, pinching the seams closed with wet fingers.
Place on a parchment-lined baking sheet, spaced 2” apart. Cover loosely.
Let rise 2-3 hours, or until noticeably puffy.
Preheat oven to 375ºF (190ºC).
Bake for 20-25 minutes or until the tops are golden brown and the internal temperature reads 190-210ºF (88-93ºC).
Transfer the sourdough hot dog buns to a wire rack to cool before slicing.