Sprinkle brisket with salt and pepper. Heat oil in a wide, heavy pot over high heat. Add brisket to the pot and brown all over, about 5 minutes per side. When the beef has browned, remove pot from heat. Transfer beef to slow cooker set on high.
Peel and slice onions and add to the original beef pot. Put the pot on medium heat and cook onions 15 minutes or so, stirring often. They should be limp and brown. Scrape into the slow cooker. Combine tomato paste and broth in the pan, stir to scrape up any bits from the pot and pour all into the slow cooker. If desired, lift up the meat, add the carrots and replace the meat. Cook on high heat for 6 to 8 hours, until meat is tender.
The meat will improve in flavor if you let it rest overnight in the refrigerator. Reduce sauce or thicken with fl our if desired.