Combine flour, brown sugar, salt, and yeast in the bowl of a stand mixer fitted with a dough hook.
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 knead the dough for 2-3 minutes on low speed until it becomes smooth, elastic, and slightly tacky (but not sticky) to the touch.
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 the dough has doubled, punch it down, then cover and letrest 10 minutes.
Divide the dough into eight equal pieces. Use a kitchen scale if you want to be precise.
To shape the dough into rounds, gently flatten a piece of dough on a clean, unfloured surface. Then fold the top of the dough down over the middle, rotate the dough 45° and repeat. Keep going all the way around the dough, folding the top edge down over the middle until you have a smooth surface against your counter and the "seam" side facing up. Flip the dough over and cup your hand gently around it with your pinkie against the counter. Slide your hand toward your body to push 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 with the rest of the dough. Cover with a damp paper towel and let rest 10 minutes.
Starting with the dough round that had the most time to rest, 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.Repeat with the rest of the dough rounds. 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.Preheat the oven to 425°F with rack in the middle of the oven.
Working in batches, boil the bagels 1 minute per side. Remove the bagels to a parchment or silicone mat lined sheet pan dusted lightly with semolina flour.
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.
If the dough seems dry during the kneading step, add water ½ tablespoon at a time, kneading between additions. If the dough seems too wet, add flour 2 tablespoon at a time, kneading between additions. This is a low hydration dough, so try not to add water unless you really think you need to.
If kneading the dough by hand, it will take about 8-10 minutes of hand kneading on a lightly floured countertop for the dough to come together.
Store bagels in an airtight bag or container. They'll stay good for 3-4 days. You can also slice them almost all the way through and freeze them.