The sausages become very tender over eight hours, and infuse the lightly sweetened cabbage with the juices they exude in the process.
Put the cabbage in the slow cooker and sprinkle with the brown sugar and salt.
In a large skillet or sauté pan, heat the oil over medium-high heat. When it shimmers, add the sausages and cook until browned, about 2 minutes on each side. Arrange the sausages over the cabbage in the cooker. Add the onion to the skillet over medium heat and cook for 1 minute, then pour in the beer, stirring to scrape up any browned bits; pour the liquid into the cooker. Cover and cook on low for 8 hours.
Using tongs or a slotted spoon, transfer the sausages, onion, and cabbage to a platter and serve, spooning some of the cooking liquid over the sausages and vegetables.
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