In a bowl, combine the water and sugar, stirring until the sugar has dissolved. Add the yeast and let it sit for about 10 minutes.
Place the multigrain bread flour and salt in the bowl of a stand mixer. Once the yeast mixture has had time to activate, add it to the flour and mix until a crumbly dough forms. Knead on low speed for about 10 minutes, until you have a smooth, elastic dough.
Remove the dough from the bowl and shape it into a ball. Lightly grease the bowl with oil and return the dough to it. Cover with plastic wrap and let it rise in a warm place for 2 hours, or until doubled in size.
After rising, remove the dough from the bowl and divide it into two equal portions. Divide each portion into four pieces, giving you eight pieces total. Roll each piece into a ball, flatten it slightly, and use your thumbs to poke a hole through the center. Gently stretch the hole until it resembles the shape of a bagel. Let the shaped bagels rest for 10 minutes.
While the bagels are resting, bring a large pot of water to a boil. Carefully lower the bagels into the boiling water, one at a time or two at a time if your pot is large enough. Boil for 45 seconds to 1 minute, then flip and boil the other side for the same amount of time. Once both sides have been boiled, transfer the bagels to a baking sheet lined with parchment paper.
Brush the wet bagels with the beaten egg and sprinkle with sesame seeds.
Bake in a preheated oven at 425°F/220°C (conventional oven) for about 20 minutes, or until golden brown.
Let the multigrain bagels cool on a wire rack, or serve them while they're still warm.