To make the cheddar beer cheese spread, add cheddar cheese, garlic, green onion, mustard and horseradish to the bowl of a food processor fitted with a metal blade.
Process until mixture forms a thick paste. With the machine running, add beer, processing until creamy.
Add additional beer as needed, to get desired consistency. Transfer spread to a bowl; cover and refrigerate 2 hours to allow flavors to blend.
To make the beer braised onions melt butter in medium skillet. Add onion and cook until translucent but not browned. Sprinkle with sugar and add beer.
Stir and reduce heat to low, allowing onions to braise until beer has evaporated, about 15 to 20 minutes. Check onions frequently, adding additional water or beer if they dry out too quickly.
To assemble the burgers, cook burger to desired doneness and top with cheddar beer cheese spread. Toast pretzel bun, if desired. Top bottom of bun with burger; top with stout-braised onions and bacon. Top with bun.