First, Place all of your ingredients in the order listed above, (Water, milk, egg, butter, sugar, flour, salt and yeast) Set the bread machine to the “dough” setting and press start. The dough setting will mix the dough and allow for a dough rise. When the machine cycle is done, punch down the dough in the pan and let it rest for 10 minutes. Remove dough from bread machine and divide it into eight equal pieces, then shape each piece of dough into a small ball by rolling it gently in a circular motion on the (un-floured) counter.
If you are making the jalapeño cheddar flavored bagels, knead diced jalapeños and shredded cheddar cheese into your desired amount of dough pieces then proceed with the steps below.
Once you form the dough balls, coat your finger in flour and press into the center of each dough ball to form a ring. Stick both pointer fingers inside the hole and swirl your bagel dough around your fingers, gently stretching the ring into a bagel shape. Cover bagels with a towel.
While bagels are resting, preheat oven to 350 and bring a large saucepan of water to a boil.
Place 2-3 bagels into the boiling water with a slotted spoon (or by gently lowering them into the water). Let bagels sit in boiling water for 1 minute, then flip and allow to sit in water for an additional 1 minute (the longer they are in the water, the chewier the texture will be).
Remove from water and place on baking sheet lined with a silicone mat or parchment paper. Using a pastry brush, brush bagels with egg wash (1 beaten egg and 1 tablespoon water). Add desired toppings. To make multiple flavors of bagels in one batch, pour your toppings onto different plates. Dip each bagel into the different topping to create a multiple flavor batch of homemade bagels, or leave your bagels plain, if desired.
Bake at 350 for 20-25 minutes or until golden. Serve with butter, cream cheese, or whatever you may like. Enjoy!