In a small bowl, combine the warm water, active dry yeast, and 1g of the sugar. Stir gently and let sit for 5 to 10 minutes until foamy. If the mixture does not foam, your yeast is no longer active and you should start with a fresh packet before continuing.
In a medium bowl, whisk together the bread flour, salt, and remaining 15g of sugar until evenly combined.
If the Greek yogurt is coming straight from the fridge, microwave it for about 30 seconds and stir well before using. You want it warm to the touch but not hot. Add the warmed Greek yogurt, egg, and bloomed yeast mixture to the bowl of a stand mixer fitted with the dough hook. Mix briefly on low speed to combine.
Add the dry ingredients to the wet ingredients and mix on low speed until a shaggy dough forms, about 1 minute.
With the mixer running on low, add the softened butter one tablespoon at a time, waiting for each addition to be fully incorporated before adding the next. This slow addition prevents the dough from becoming greasy and allows the flour to absorb the fat properly.
Once all the butter is incorporated, increase the mixer to medium speed and knead for 6 minutes until the dough is smooth, elastic, and pulls away cleanly from the sides of the bowl. The dough should be tacky but not sticky.
Shape the dough into a smooth ball and place it in a lightly oiled bowl. Cover with plastic wrap or a damp towel and let it rise at room temperature for 1 to 1.5 hours, or until doubled in size.
While the dough rises, grease all 10 wells of the pan well, even if it is nonstick. Pat each hot dog dry and set aside.
Turn the dough out onto a lightly floured surface and weigh the total dough. Divide into 10 equal pieces of approximately 65 grams each using a bench scraper.
Roll each piece across the counter into a log roughly the length of a hot dog. Start in the middle and pull outward as you roll back and forth. Let the pieces rest 5 to 6 minutes so the gluten relaxes.
Pat a piece of dough flat into an oval. Lay a hot dog in the center and press the sides of the dough all the way down to the bottom of the dog so it wraps around, leaving the bottom part of the hotdog open (which will be the top after baking and flipping). Place it dog side down into a well of the pan, smooth dough on the bottom. Repeat with the remaining dogs and dough.
Cover loosely and proof for 45 minutes to 1 hour, until noticeably puffy. Preheat the oven to 375 degrees F during the last 20 minutes of proofing.
Whisk the egg with the water until smooth. Gently brush the tops with the egg wash with a piece of paper towel or pastry brush.
Bake at 375 degrees F for 25 to 30 minutes, until deep golden brown. The buns are done when the bread reads 190 to 195 degrees F internal and the hot dogs are heated through.
Let the pan rest for five minutes
Flip the buns out of the tray.
Brush the hot buns immediately with melted butter.
Let the buns cool about 15 minutes before separating them, since they bake as one connected slab and need to set before you pull them apart cleanly.