Bring about 2cm (1in) salted water to the boil in a large saucepan.
Slice cabbage, leaving as chunky of fine as you like. Simmer cabbage, covered for 3-4 minutes or until cabbage is softened but still has some bite. You want it to be ‘al dente’ like cooked pasta. Drain.
Return cabbage to the pan and stir in tuna, pesto and a good glug of extra virgin olive oil. Taste and season with salt and more pesto as needed. Serve in big bowls with or without the shaved parmesan.