Combine the first four ingredients together in a bowl, mixing until well combined. Set aside.
In another bowl, microwave together the cheeses until melted, about 30-45 seconds. Stir, then microwave again for another 20-30 seconds if needed until mixture is smooth.
Add in the flour mixture along with 3 of the eggs. Mix until smooth. If mixture is getting stiff, return to microwave for about 20 seconds.
Use wet hands to shape 10-12 bagels, and place on lined baking sheet.
In a small dish gently whisk remaining egg, then brush on tops of each bagel. Sprinkle with Everything seasoning (or other seasoning of choice), then bake at 425°F for 12-14 minutes.
Notes
Originally I made 6 large bagels with this recipe, however because the dough is so rich, I could only ever finish 1/2 a bagel when I made them that large. So I recommend dividing the dough into 10-12 bagels.