Line two rimmed baking sheets with parchment paper (cut them to fit) or just grease your pans.
In a medium or large mixing bowl, crack the egg and beat it with a fork.
Add all other ingredients to the bowl and stir with a wooden spoon to combine. If the dough is very sticky, wet, and not coming together, add a couple tablespoons of flour at a time until the dough comes together.
Sprinkle some flour on a clean counter or work surface and place your dough onto the surface. Knead the dough by hand for five minutes. It should be smooth and a little tacky, but not sticky. Choose either the Roll & Cut Method or the Dough Balls Method for shaping the buns.
Roll & Cut Method (for uniform results, but more dense texture)
On a floured surface, use a rolling pin to roll out the dough until it is ½-inch thick.
Cut out circles of dough using a glass or biscuit cutter, or for larger buns as pictured, use a plastic quart container with a 4 ½-inch-wide opening as a cutter. Place the circles of dough on your prepared pans, spacing them apart so the dough circles don't touch.
Once you've cut all the circles of dough you can fit from the first rolling, you'll need to gather the dough in a ball again, flatten it and roll it ½-inch thick. Then, proceed to cut out more circles of dough. Gather the scraps of dough into a ball and roll again as needed. I get 9 buns cut 4 ½-inches wide. Proceed to the Cover, Rest & Bake instructions further down.
Dough Balls Method (for fluffier texture, but the buns won't all look the same)
Please read this section carefully before starting. Use a bench scraper or straight-edged knife to separate the dough into nine, approximately even pieces. (Cut the dough into three equal parts, then cut each of those parts into three equal pieces.) IT IS IMPORTANT NOT TO ADD DOUGH TO ANY OF THE PIECES OR YOU'LL GET CRACKS WHEN THE BUNS BAKE.
Roll each piece of dough into a smooth ball. I roll it on the counter, rather than in my hands. Place the dough balls on your baking pans, spaced apart so they don't touch. Gently press down each ball with the palm of your hand to make the shape a little flatter.
Cover, Rest & Bake
Loosely cover the pans of shaped hamburger bun dough. (Use clean, lint-free kitchen towels or greased plastic wrap to cover them.) Preheat your oven to 425 degrees F.
Let the covered dough rest for 10 minutes.
Bake, one pan per oven, on the center rack, for 10 minutes or until the buns are golden on top.
Let the hamburger buns cool on a wire rack before slicing them in half to serve. Freeze leftover buns in a freezer bag and thaw them before use.
Using only all-purpose flour will give you a fluffier bun. I use one cup of white whole wheat flour, and 2 ½ cups of all-purpose flour, and the result is a bit more dense.
If it is humid and your dough is sticky and not coming together, you will need to add more flour. Add a couple tablespoons at a time, until the dough comes together.