In the bowl of a stand mixer fitted with a dough hook, mix the flour, sugar, yeast and salt on low speed for 30 seconds to combine.
In a seperate jug whisk together the milk and eggs, then pour most of it into the flour mixture. (Hold back a couple of tablespoons of liquid in case you don't need all of it.)
Switch the speed to medium and knead the dough until it comes together into a soft ball. If it seems a bit dry after a few minutes, add the remaining liquid, about a teaspoon at a time, until it comes together around the hook but still sticks quite a bit to the bottom of the bowl.
Add in the softened butter, one slice at a time and knead for 15 seconds between each addition. Continue to knead the dough for about 8 more minutes. The dough should be very soft.
Turn out the dough onto a lightly floured surface and shape into a smooth ball. Place it into a greased bowl, cover the bowl with plastic wrap (or a shower cap) and keep in a warm place to proof for 1 ½ - 2 hours, until the dough no longer springs back when poked.
Shape and Bake the Hot Dog Buns Divide the dough into 6 equally sized pieces (about 3 ½ oz/100 g). Shape each piece into a smooth roll (see video above), then elongate into a 6-inch (15 cm) log. Pinch the seams of the dough together well so the bun doesn't pop open while baking. The dough should be smooth and crease-free (Shaping the dough into a smooth ball is very important. See my How to Shape Rolls video!)
Place the hot dog buns on a lined baking sheet, seam-side down, cover with plastic wrap and proof for 45 to 60 minutes, until puffed slightly and the dough no longer springs back when poked. Once proofed, preheat the oven to 375°F (190°C), then brush each bun with egg wash.
Bake for 20-22 minutes or until they are golden brown. Remove from the oven and cover with a clean, dry tea towel as they cool to soften the crust. Store completely cooled buns in an air-tight container in the refrigerator or at room temperature for up to 2 days. For longer storage, freeze for up to 6 weeks. Defrost at room temperature for about 1 hour.