Line a shallow baking tray about 20 cm square with non-stick baking parchment. Bring the water or vegetable stock to the boil with some salt and reduce the heat to produce a rapid simmer. Whisk in the polenta in a continuous stream, then change your whisk for a spoon and cook for 1 minute, stirring all the time. Add the butter and sesame seeds and continue to cook until the butter is absorbed. Turn the polenta out on the tray and, using a palette knife, spread it out to an even thickness of about 7 mm. Leave to cool completely, then cut into whatever shapes you like, either with a knife or a cookie cutter.
Preheat a warm oven. Heat 3 cm of oil in a pan and fry the polenta shapes in batches, until golden and crisp on both sides, then place in the warm oven to keep hot.
Place the olive oil in a moderately hot pan and cook the anchovies, mashing them with the back of a spoon as they cook, until they are almost dissolved. Add the diced courgette and continue to cook and stir until they have just begun to soften. Add the lemon zest and juice, and cook for another minute. Take off the heat and leave to cool a little, then mix in the spring onion, cheese and basil.
Divide the mixture between the polenta shapes and serve while still warm.