Pat shanks dry and season with salt and pepper. Heat oil in a large skillet over medium-high heat and brown shanks for about 4 minutes per side, then transfer to a 6-quart slow cooker.
Add onion to the skillet and cook for 4 minutes. Stir in garlic for 30 seconds, pour in 1 cup of the measured broth and scrape up the browned bits.
Pour skillet liquid into the cooker and add remaining broth, carrots, celery, tomatoes and bay leaf. Cover and cook on low for 8 to 9 hours until beef is fully fork-tender and well above 145°F.
Lift shanks onto a clean plate. Remove and discard every bone and any tough connective pieces, pull the meat into bite-size shreds and return it to the broth; discard the bay leaf and skim excess fat.
Stir in green beans, corn and peas. Cover and cook on high for 25 to 30 minutes until the vegetables are hot and tender.
Stir in red wine vinegar and divide among 6 bowls, checking once more for stray bone fragments as you ladle.