Cut the core from the cabbage. Fill a large pot 1/3 way full with water and bring to a boil. Place cabbage in boiling water, cover and cook for about 10 minutes. Remove cabbage and let it cool enough to handle it easily. Carefully separate about 12 leaves, drain and set aside.
Add oil to a large skillet over medium heat. add half the onion and saute until it begins to get soft. Remove from heat; set aside.
Mix the beef, pork, cooked rice, garlic, cooked onions, salt and pepper in a large bowl.
Remove the thick rib from each cabbage leaf and place about 1/2 cup of filling in each leaf. Fold in sides then roll up so meat is entirely covered. Repeat with each leaf.
Mix the sauce ingredients in a small bowl.
In the same skillet you cooked the onions, brown the cabbage rolls. Top with remaining onion and shredded carrots. Pour sauce over rolls, bring to boil. Cover and reduce heat to medium-low and cook for about 45 minutes.
Remove cabbage rolls to a platter and keep warm. Reduce sauce by half over medium-high heat, about 5 minutes. Pour over cabbage rolls.