Rinse ribs and pat dry. In a paper bag, combine flour with 1 tablespoon each salt and pepper. Drop ribs into bag and shake to coat. Lift ribs out, shaking off excess flour mixture.
In a heavy 6-quart pan over medium-high heat, melt butter with olive oil. Working in batches, add ribs in a single layer and turn to brown on all sides, about 5 minutes total per batch. With tongs, transfer ribs to a bowl. Discard all but about 2 tablespoons fat in pan.
Reduce heat to medium and add onions and garlic to pan; stir often until onions are limp, about 6 minutes. Stir in wine, mustard, and apricots, then return short ribs to pan. Cover and bring to a simmer, then reduce heat to maintain a low simmer and cook, turning ribs once or twice to submerge meat, until very tender when pierced, 2 to 2 1/2 hours.
With tongs or a slotted spoon, transfer short ribs to individual wide, shallow bowls or a serving bowl. Skim off and discard any fat from pan juices. Boil juices over high heat until reduced to about 3 cups. Season to taste with more salt and pepper. Pour juices over ribs.
Wine Pairing: A hearty Cabernet Sauvignon with meaty tannins matches the richness of the dish and brings the sauce's flavors to life.