Bloom yeast: Stir warm water and sugar in a large bowl, sprinkle in yeast, and let sit until foamy, 5–10 minutes.
Mix dough: Whisk in melted butter and salt. Add flour gradually, stirring until a shaggy dough forms.
Knead on a lightly floured surface until smooth and elastic, 5–7 minutes. Cover and rest 10 minutes.
Let rise: Place dough in a lightly greased bowl, cover, and rise in a warm spot until puffy, about 30 minutes.
Season cheese: Toss cheddar with garlic powder and paprika; set aside.
Preheat oven to 425°F. Line two baking sheets with parchment and lightly oil them.
Prepare bath: Bring 10 cups water to a gentle boil in a wide pot. Stir in baking soda carefully.
Shape ropes: Divide dough into 6 pieces. Roll each into ¾-inch-thick ropes and flatten slightly.
Fill and cut: Sprinkle a narrow line of cheese along each rope, pinch seams to seal, then cut into 1-inch bites.
Boil: Drop bites in batches into the baking soda water for 30 seconds, turning once. Drain well.
Top and bake: Arrange on sheets, sprinkle with coarse salt, and bake until deep golden, 10–12 minutes.
Finish: Brush hot pretzel bites with melted butter. Cool 5 minutes and serve warm.