Heat a 12-inch cast-iron skillet over medium-high heat.
Reduce heat to medium; add butter, stirring until butter is melted.
Add onion, garlic, ¼ teaspoon salt, and pepper; cook, stirring frequently, until onion is golden brown and softened, about 10 minutes.
(Onion will not be tender at this point.)
Stir in vinegar and brown sugar.
Reduce heat to medium-low; cover and cook, stirring occasionally, until onion is very soft, about 1 hour and 15 minutes.
Stir remaining ¼ teaspoon salt into onion mixture.
Let cool.
In the work bowl of a food processor, pulse onion mixture until finely chopped.
Serve warm or at room temperature with blue cheese and crackers.
Store in an airtight container in the refrigerator for up to 1 week.
