Mix flour, salt, and sugar together in the bowl of a stand mixer fitted with a dough hook. Make a well in the middle of the dry ingredients and sprinkle in the yeast. With the mixer running on the slowest speed, pour the water into the center of the well. Continue on low, pausing occasionally to push more flour into the center of the well with a spatula. When the dough starts coming together but there's still a decent amount of dry flour in the bottom of the bowl, add the mashed garlic and fennel. Once the dough has incorporated and no loose flour remains, Increase mixer speed to low-medium and knead for 6-8 minutes.If the dough looks dry, you may need up to an additional ⅛-¼ cup of water. Add it ½ tablespoon at a time, kneading well between additions to give the water time to absorb. Resist adding water unless absolutely necessary. If the dough seems too wet, add more flour 2-3 tablespoon at a time, kneading well between additions. You’re looking for a dough that is smooth and just slightly tacky to the touch. It shouldn’t cling to your hands or the bowl. Shape the dough into a smooth round ball, and place it in a lightly oiled bowl (turn it once in the bowl to coat it with oil) and let it rise for an hour in a warm spot (70-72°F) until doubled in size.
Gently punch the dough down, knocking the air bubbles out of it. Let it rest for 10 minutes.
Preheat your oven to 425°F and fill a wide, shallow skillet with water about 2-3" deep. Bring to a boil, then reduce to a simmer.
Divide the dough into eight equal pieces (use a kitchen scale for precision). Gently flatten each piece of dough against a clean, un-floured countertop, then fold and tuck the edges up into the middle, pinching them together to form a smooth surface on the underside of the dough against the counter. Flip the dough over so the seam is underneath. Cup your hand around it, pinkie against the counter, and drag your hand towards your body. Rotate the dough 90 degrees and repeat as needed to form a ball. Starting with the first dough ball you shaped, coat your thumb in flour and poke it through the seam-side of the dough, pushing any extra edges into the middle. Slide your other thumb in and gently stretch the dough out until the hole in the middle is at least the same width as the sides. The hole will close as the dough rests and will close again as it boils and bakes, so better to stretch it wider here.NOTE: If you like a bigger hole in your bagel, stretch the dough out again before boiling. Cover the shaped bagels with a damp paper towel or clean dish cloth and let them rest for 10 minutes.
Whisk together egg and 1 teaspoon water to make an egg wash while the bagels rest. Set aside.
Gently slide your shaped bagels into the pot of boiling water, top-side down. Boil for 1 minute per side. You can do this in batches of 2-4 at a time, depending on how big your pot is. The bagels will expand as they boil so don’t crowd them. Remove the shaped bagels from the water with a wire spider or slotted spoon and transfer them to a silicone baking mat lined baking sheet. Brush each bagel with the egg wash making sure to get the sides and centers. If you want to top your bagels with additional ground fennel seeds, now is the time to do so.
Bake them for 20 minutes, until golden brown on top.
Remove the bagels from the oven and transfer immediately to cool on a wire rack. Try to wait at least 15-20 minutes before cutting them open.