Combine flour, brown sugar, salt, and yeast in the bowl of a stand mixer fitted with a dough hook.
In a separate bowl, whisk eggs, egg yolks, and water together until well combined. Reserve one egg white to use as an egg wash later.
Make a well in the center of the dry mixture. Pour the eggs and water into the center of the well. Run the mixer with the dough hook on a low-medium speed, scraping down the sides of the bowl as needed until a shaggy dough forms. Then increase the speed to medium and knead the dough for 2-3 minutes until it becomes smooth, elastic, and slightly tacky (but not sticky) to the touch. If the dough is clinging to the walls of the bowl, dust in more flour. If the dough seems dry, drizzle in water ½ teaspoon at a time letting the mixer run for at least 30 seconds between additions.
Shape the dough into a ball and let it rest, covered, in a lightly greased bowl for 1 hour or until doubled in size. When you press a finger into it, the indentation should spring back slightly but not all the way. If the indentation fills in immediately and completely, let rise for an additional 10-20 minutes. Gently deflate the dough in the container, then cover and let rest for 10 minutes.
Divide the dough into eight equal pieces, using a kitchen scale to be precise.
To shape the dough into rounds, gently flatten a piece of dough on a clean, unfloured surface. Fold the edges of the dough up over the center, pinching together at the top. Flip the dough ball over so that you have a smooth surface facing up, and the pinched together "seam" is on the counter.Gently cup your hand around the dough with your pinkie against the counter. Slide your hand in a straight line toward your body to pull the dough ball closer to you. This will increase the surface tension on top of the dough and shape it into an oval. Rotate the dough 90° and repeat the sliding motion to turn the oval into a circle. Repeat as needed until the dough ball feels tight and smooth. Cover bagel balls with a damp paper towel and let rest 10 minutes. Dip your thumb in flour and poke it through the bottom seam of the dough and out the other side. Slide your other thumb in next to it and stretch the dough by squeezing and rotating it through your hands until the center hole is at least the same width as the outside of the bagel. Don't squeeze too hard or tear the dough; gentle pressure as you rotate the bagel through your hands will slowly stretch it just fine.Cover the shaped bagels with a damp paper towel and let rest 10 minutes.
While the bagels rest, fill a large, high-sided skillet about halfway with water and preheat the oven to 425°F with rack in the middle of the oven. Line a sheet pan with parchment or a silicone mat.
Working in batches, boil the bagels 1 minute per side. Remove the boiled bagels to the sheet pan.
Whisk together one of the leftover egg whites with a splash of water and pinch of salt to make an egg wash. Brush the boiled bagels with the egg wash and sprinkle with your preferred toppings.
Bake the bagels for 20 minutes. Remove from the oven and let cool on a wire rack.