In a large bowl or stand mixer, combine lukewarm water, yeast, and sugar. Let stand for 5 minutes.

Add salt, cornmeal, and 2 cups of flour. Mix to combine.

Add another cup of flour and mix in well. Gradually add the last cup of flour until the dough wraps around the kneading hook and cleans the bowl.

Transfer the dough to a lightly floured surface. Knead in jalapeño pepper and cheddar cheese until evenly distributed.

Place dough in a greased bowl, cover with plastic wrap, and let rise until doubled, about 45 minutes to 1 hour.

Prepare a baking sheet with parchment paper and sprinkle with cornmeal. Set aside.

Deflate the dough and divide into 8 equal pieces. Shape each piece into a ball, then poke a hole in the middle and stretch into a bagel shape.

Place bagels on the baking sheet and cover with a clean tea towel. Let rise for 30 minutes.

Bring a large pot of water to a boil and preheat the oven to 400°F.

When bagels are ready, add baking soda to boiling water. Boil 2 to 3 bagels for 1 minute on each side. Remove and let excess water drip before placing back on the baking sheet. Repeat with remaining bagels.

Brush bagels with egg wash and sprinkle with additional cheddar cheese.

Bake in preheated oven for about 22-23 minutes, checking at 20 minutes. Cover loosely with aluminum foil if necessary to prevent over-browning. Bagels should be deep golden and sound hollow when tapped.

Transfer bagels to a cooling rack to cool completely.