Yes, I've made homemade bagels, bialys, bagel dogs and pitas for a long time. The trick with bagels is you need to mix, knead, rise, shape and then boil them before baking. Bagel dogs I also boil before baking, bialys are not boiled.
Here is my basic recipe
6 Âľ cups flour, 1/2 bread 1/2 all purpose
1 â…“ cups warm water
1 ounce fresh yeast or 1 T dried yeast
ÂĽ cup sugar
1 ½ teaspoons salt
3 tablespoons oil
2 eggs
Mix all ingredients in food processor or stand mixer until dough is elastic. Raise 1 hour. Punch down, shape and boil 2 minutes. Glaze with beaten egg white and top with favorite toppings. Onions, poppy seeds ect.
Bake 500 degrees until brown.
The rules aren't strict, just use basic bread making skills and substitute the correct method if mixing with a stand mixer or by hand.
I can't find my pita recipe right now, but it's really easy to make.
I realized I forgot to add part of the process. The raise one hour is assuming you are allowing a slow rise in the fridge for 9-12 hours ( can go up to 24) and then rising 1 hour after removing from the fridge. Alternatively, you can rise on the counter until doubled.