Add warm water to the bowl of your mixer and sprinkle the yeast over top. Let sit 10 minutes, until frothy.
Add the, oil, 1 egg, and sugar to the bowl.
Add the flour, salt, cheese and jalapeno peppers and mix on medium-low speed with the hook until the dough holds together, about 2 minutes. The dough should clean the sides of the bowl. If not, add additional flour by the tablespoon.
Increase speed to medium and knead the dough for 5 minutes.
Transfer the dough to a large bowl that has been lightly greased with oil.
Cover it with a towel and let rise in a warm place until doubled in bulk, about 1 1/2 hours.
Line a baking sheet with parchment.
Turn out the dough onto a lightly floured surface.
Divide the dough in half and each half into 4-6 pieces (depending on the size of your burgers, typically about 8-9 buns per batch).
Roll into rounds and flatten with the palm of your hand onto the baking sheet; placing buns about 1/2-1 inch apart.
Cover with a damp towel and let them rise for 20 minutes + oven preheating time.
Preheat oven to 350.
Whisk the remaining egg with 1 tablespoon water.
Brush the buns with the egg wash and bake for 20 to 30 minutes, until golden brown. Let cool minutes before slicing and serving.
Store leftovers in a zip-top bag at room temperature. Freezes well.