Place warm water in the bowl of a stand mixer fitted with a dough hook OR a large bowl. Dissolve yeast in water and let stand 5 minutes until foamy.
Mix brown sugar and salt into yeast mixture until smooth. Add 1-1/4 cups flour to bowl and mix on low speed OR with a wooden spoon if making dough by hand until dough pulls away from sides of bowl, about 2 minutes of mixing (if dough seems a bit wet, add additional flour by the tablespoon until dough pulls away from sides of bowl)
If making dough in stand mixer, knead dough in mixer on low speed 5-7 minutes until dough is stretchy and elastic. If making dough by hand, turn dough onto a lightly floured surface and knead by hand until dough is stretchy and elastic, about 10 minutes of kneading.
Transfer dough to a cutting board and cover with plastic wrap or a kitchen towel. Allow to rest 20 minutes.
Preheat oven to 425F and line a large sheet pan with parchment paper. Set aside.
Divide dough into 8 pieces (we recommend using a kitchen scale to make sure they are all the same size) and roll each piece into a ball. Flatten each ball, place a square of cheddar cheese in center, and wrap dough around cheese, pinching edges to seal and rolling each dough portion into a smooth ball.
Bring water to a boil in a large pot or Dutch oven. Dissolve baking soda in boiling water (careful, it will bubble a lot). Add pretzel bites to boiling water and allow to boil 1 minute. Transfer pretzel bites from boiling water to prepared sheet pan using a slotted spoon.
Brush pretzel bites with egg white wash and immediately sprinkle with sea salt. Bake pretzel bites at 425F 12-15 minutes until a deep golden brown.
Remove pretzel bites from oven and lightly brush with melted butter. Serve pretzel bites immediately warm with your choice of dipping sauce. Enjoy!