In a food processor, add the flour, yeast, salt, and sugar. Pulse until incorporated.
While running, slowly add the warm water until the dough balls up. You may not need all the warm water.
Carefully push the dough back into the blade’s path and continue to process for an additional 30 seconds or until the dough is firm and stretchy, but not sticky nor squishy.
Gently round the dough into a ball and place in a lightly oiled bowl. Cover with a damp, warm towel for an hour or until the dough has doubled in size.
Add the barley malt syrup to about 6 quarts of water and bring to a boil.
Place a tray with about 2 quarts of water on the bottom shelf of the oven. Preheat to 425°F (220°C).
Place the dough onto a lightly floured surface and divide into 6 equal pieces (roughly 5 ounces (140 g) a piece if you use a scale). Cover the pieces with a damp towel.
One at a time, form the dough into a ball, pinching the seams together at the bottom. Place the dough seam side down and–cupping your hand over the dough–apply gentle pressure, working the ball in circles until the seams on the bottom are smoothed out. Gently reshape the edges of the ball into a circle.
Using your thumb, press down into the center of the ball to the counter. Working in a circle, carefully stretch the dough into a ring until the hole at the center is about 2 inches (5 cm) in diameter.
Place the dough rings under a damp towel and let rise for 10 minutes.
Working in batches of two, place the bagels in the boiling water/syrup. They should float at the surface. Let them boil for 30 seconds, then flip and boil for an additional 30 seconds.
Remove the bagels from the water/syrup and gently pat dry with a clean towel.
Brush the tops with the beaten egg. Optionally, if you want toppings, dip the egg-washed side of the bagel into the mixture of your choice.
Place the bagels on a parchment paper-covered baking sheet. Bake until golden brown, about 15 to 20 minutes.