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In the bowl of your electric mixer, combine warm water, active dry yeast, and sugar. Stir and let sit for 10 minutes, until it gets foamy.
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Add in flour, green chile, and salt. Mix with dough hook until dough is formed, then continue to mix for 5 minutes. If the dough is too sticky, add in more flour, or if dough is too dry, add in a little bit of water, until dough is elastic and manageable. Drape towel over the mixing bowl and let dough rise for one hour.
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Turn dough onto floured surface and separate into 8 equal-sized balls. Place onto greased baking sheet, cover, and let rise for 15 minutes. Meanwhile bring a large pot of water to a boil.
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Poke holes into the center of each ball, forming the center hole of the bagel. Repeat with all bagels, then cover and let rise another 15 minutes. Meanwhile, preheat oven to 415 degrees .
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Place bagels, two at a time, into the boiling water. Boil bagels for 2 minutes on each side, then remove, drain, and place back onto baking sheet. Repeat for all remaining bagels.
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Top bagels with additional green chile (optional) and a handful of cheese to each bagel, then bake in preheated oven for 25 minutes, until cheese is starting to brown.