Place the warm water in a large bowl and sprinkle with the yeast. Let stand for a few minutes, until it starts to get foamy. Add 2 cups of the flour and the salt and stir until the dough comes together. Continue to stir, adding 1/2 cup or so more flour, until the dough comes together but is still tacky. Knead or stir with the dough hook on your stand mixer for 5 minutes, or until smooth. Leave in the bowl, covered with a tea towel, for an hour or two, until doubled in bulk.
Meanwhile, set a small skillet over medium heat, add a drizzle of oil along with the butter and when the foam subsides, cook the onion for 15 minutes or so, stirring often, until it’s soft and pale golden. If you like, stir in some poppyseed.
Divide the dough into pieces (about 8 — but make any size you like) and on a parchment-lined baking sheet, shape each into a circle with a deep indent in the middle, like a small pizza with a thickish edge. (It will puff up as it bakes.) Put a spoonful of caramelized onions in the middle of each, and if you like, grate a bit of cheese overtop. Let them sit as you preheat the oven to 450F. Bake for about 20 minutes, or until puffed and deep golden. Makes about 8 bialys.