In the bowl of a stand mixer fitted with the dough hook or in a large bowl by hand, combine the warm water, sugar, oil, yeast, egg, salt and 2 cups of the flour.
Start the mixer on low speed. Continue adding flour gradually until the dough forms a soft ball and pulls away from the sides/center of the bowl. Knead for five minutes (if the dough starts clinging to the sides again during kneading, add a few tablespoons flour at a time until it clears the sides of the bowl). The dough should feel soft and sightly tacky to the touch but shouldn't leave a lot of dough residue on your fingers.
Cover the bowl and let the dough rise until doubled in a warm, draft-free spot, 45 to 60 minutes.
Line two half sheet pans with parchment paper. Lightly grease with cooking spray and set aside.
Lightly punch down the dough and turn it out onto a lightly greased countertop.
Divide the dough into 10 equal pieces (about 4.5/4.7 ounces each). Roll each piece into a ball and place the rolls on the prepared pans - five rolls for each pan spaced several inches apart.
Lightly press each ball of dough into a thick disc about 1/2-inch tall and 2 to 2 1/2 inches across. If the dough springs back after being lightly pressed, let it rest for 5 to 10 minutes to help the gluten relax and press again.
Cover and let rise in a warm, draft-free spot until noticeably puffy, about 45 minutes.
Preheat oven to 375 degrees.
For the egg wash, whisk together the egg and water. Lightly brush (so as not to deflate the risen buns) the top and sides of each bun with the egg wash. Sprinkle the top with sesame seeds, if desired.
Bake for 18 minutes until golden.
Notes
Yeast:to sub in active dry yeast for the instant yeast, add the water, sugar, oil, and active dry yeast to the mixer. Mix briefly. Let the mixture rest until the yeast is foamy and bubbling before proceeding with the recipe. Egg:to make the buns egg-free, omit the egg and add 2 to 3 tablespoons additional water (and omit the egg wash). The buns will be slightly less sturdy without the egg, but they will still work great for burgers.Flour:you can sub in half whole wheat flour (preferably white wheat variety). Knead for 1-2 minutes longer. Judge the exact amount of flour by the look and feel of the dough. The flour amount may vary slightly based on elevation, humidity, how the flour is measured. Size of Buns:These buns are about 3.75 to 4 inches in diameter once baked (the size is variable depending on how flat the buns are pressed prior to rising and baking).