In a small bowl, combine the warm water, active dry yeast, and 2g 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 If the mixture does not foam, your yeast is no longer active and you should start with a fresh packet before continuing.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 25g sugar until evenly combined.
If the Greek yogurt is coming straight from the fridge, microwave it for about 40 seconds and stir well before using. You want it semi warm to the touch but not hot. You're just taking the chill off.
Add the warmed Greek yogurt, eggs, and bloomed yeast mixture to the bowl of a stand mixer fitted with the dough hook. Mix briefly on low speed to combined.
Add the dry ingredients to the wet ingredients and mix on low speed until a shaggy dough forms, about 2 minutes.
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 to 8 minutes until the dough is smooth, elastic, and pulls away cleanly from the sides of the bowl. The dough should be soft but not sticky. If it is really sticking to the bowl after the full knead, add flour 10 grams at a time until it pulls away.
Shape the dough into a smooth ball and place it in a lightly oiled bowl. Cover with plastic wrap or a lid and let it rise at room temperature for 1 to 1.5 hours, or until doubled in size.
Turn the dough out onto a lightly floured surface and weigh the total dough. Divide into 12-13 equal pieces of approximately 100-110 grams each using a bench scraper.
Roll each piece into a tight, smooth ball by tucking the edges underneath and using your hand cupped around the dough ball in a "C" shape. moving in a counter clockwise circle, creating surface tension on top. Flatten each ball firmly with your palm to about 3 to 3.5 inches wide. Flattening before the proof is what gives you a wide, flat burger bun shape rather than a tall, domed dinner roll.
Place the shaped buns on two parchment lined baking sheets, spaced at least 2 inches apart. Cover loosely with plastic wrap and let proof for 45 minutes to 1 hour, until noticeably puffy. Preheat the oven to 375 degrees F during the last 20 minutes of proofing.
Alternatively you can add them in a 4 x 3 grid so they all fit on one tray. You may need to pull them apart after baking.
Whisk the egg with the water until smooth. Gently brush each bun with the egg wash. Sprinkle with sesame seeds or everything bagel seasoning if desired.
Bake for 18 to 22 minutes, rotating the pans halfway through, until deep golden brown on top. The buns are done when the internal temperature reads 190 to 195 degrees F.
Optionally, brush the hot buns immediately with melted butter the moment they come out of the oven. This keeps the crust soft and adds a final layer of richness. Let cool on a wire rack for at least 15 minutes before slicing.