Make Pretzel Dough: Stir together warm water and yeast in a large bowl and let stand until a creamy beige foam develops on the surface, about 10 minutes. (If mixture doesn't foam, start over with new yeast.)
Meanwhile, in a smaller bowl, stir together flour, salt, and light brown sugar.
Add flour mixture and butter to yeast mixture, and stir together first with a wooden spoon and then with your hands, until it forms a dough. On a lightly floured surface, knead dough until it is smooth and elastic, 8 to 10 minutes, dusting with more flour as needed if dough sticks to work surface.
Transfer dough to a lightly oiled large bowl and turn over to coat with oil. Cover with a kitchen towel (not terry cloth) and let rise in a draft-free place at room temperature until doubled, 2 to 3 hours.
Alternatively, after kneading, let dough rest at room temperature for 15 minutes, then transfer to oiled bowl as above, and let rise in the refrigerator overnight (12 hours or up to 2 days).
Punch down dough and divide into quarters. Form each quarter into 4 balls. Roll each ball into a 12-inch-long rope. Cut each rope into six (2-inch) pieces, and transfer to wax paperlined rimmed baking sheets.
Bake Pretzels: Heat oven to 450°F with racks in upper and lower third. Well oil 2 large baking sheets (about 17- by 12- inches).
In a 4- to 5-quart saucepan, bring water and baking soda to a boil (mixture will be foamy at first, but foam will dissipate). Add 12 pretzel bites and cook 30 seconds (they will rise to the surface almost immediately), then transfer with a slotted spoon to baking sheets. Quickly sprinkle pretzel bites lightly on both sides with pretzel salt before they dry (so salt sticks). Repeat with remaining pretzel bites, returning water to a boil between batches.
Bake pretzel bites in oven, switching sheets between racks and rotating sheets halfway through baking, until deep chestnut brown, 8 to 10 minutes (watch closely toward end of baking; bites brown quickly in last few minutes). Immediately transfer pretzel bites to wire racks to cool.
Make Cheddar Dip: Purée cheese with 6 tablespoons wine or water in a food processor fitted with knife blade, stopping and scraping down sides with a spatula occasionally, until mixture is creamy and smooth, about 2 minutes. Thin to desired consistency with additional wine or water, adding it 1 tablespoon at a time. Serve immediately, or chill, covered, and return to room temperature before serving.
Cooks' Notes: You can buy pretzel salt in some specialty food stores or online. Pretzel bites are best the day they are made, but they can be made 1 day ahead and kept in a resealable plastic bag or frozen 1 month. Thaw, and rewarm, if desired, in a 350°F oven for about 5 minutes. Cheddar dip is delicious plain, but you can flavor it with your favorite prepared mustard (we used grainy Dijon). Start with 2 teaspoons, adding more to taste, as desired, and/or caraway seeds (2 to 3 teaspoons, or to taste), coarsely ground in an electric coffee/spice grinder. Caraway seeds will be even better if you lightly toast them first (before grinding) in a dry heavy skillet over medium heat, stirring, until fragrant and a shade darker, 2 to 3 minutes.