Chop cabage in quarters, and then again in large to medium sized chunks. Cut carrots into medallions 1/4-1/2" thick.
Heat olive oil in a large stock pan. Add in cabbage, carrots,and 1/2 cup water and put the lid on. Allow the cabbage to wilt before adding the remaining water (~5 minutes).
Stir in garlic paste, salt, pepper, and turmeric and cook on medium heat with the lid on for 15 minutes, stirring occasionally, before reducing to low heat.
Cut lemon in half and squeeze in the juice from both halves (~1/4 cup). Stir again, and place both halves on top of the cabbage insides down. Continue to cook on low heat (lid on) for another 45-60 minutes, depending on your desired consistency of the cabbage. You may also need to add water at some point throughout this process.
Remove and discard lemon halves. Allow to cool, and serve warm to room temperature. Pairs well with injera.