Blanch the cabbage: Bring a large pot of salted water to a boil. Meanwhile, remove 10 to 12 whole outer leaves from the cabbage by cutting them off the core. (Reserve the remaining cabbage for another use.) Add the leaves to the boiling water and blanch until wilted, 2 to 3 minutes. Drain and transfer the leaves to a paper towel–lined baking sheet; set aside.

Make the sauce: Place all of the sauce ingredients in a medium bowl and stir to combine. Pour half the sauce into a 5-quart or larger slow cooker and spread into an even layer; set the remaining sauce aside.

Make the filling: Place all the ingredients except the beef in a medium bowl and stir to combine. Add the beef and gently mix it in, making sure not to compact the meat. Divide the mixture into 8 portions (about scant ⅓ cup each), then form each portion into a ball.

Working with one leaf at a time, place a ball on the center of the leaf. Fold in the sides of the leaf tightly over the filling, completely covering the filling. Place seam-side down in the slow cooker. Repeat with the remaining leaves, overlapping 2 leaves as needed if they are too small (you will not use all the leaves), placing the rolls in a single, tight layer.

Pour the remaining sauce over the top of the stuffed cabbage so that it's completely covered. Cover and cook until the meat is cooked through and the rice and cabbage are tender, about 8 hours on the LOW setting.