Heat a dash of oil in a large frying pan, and add the diced carrot, onion, peas, and sweetcorn. Cook over a medium heat for around 10 minutes, until any excess water from the frozen veg has cooked off, and the carrot is beginning to soften. Add the mushrooms, garlic, and asparagus, and cook for a further 5 minutes. Then add the cannellini beans, and mix to combine.
Meanwhile, melt the butter in a saucepan, over a fairly low heat. Add the flour, and mix to form a roux (a thick paste). Add the milk a little at a time, stirring until smooth each time before adding more. You can find more detailed instructions for how to make a white sauce here. When you've used all the milk, and the sauce is completely smooth, add the herbs, salt, pepper, and the grated cheese. Mix over a low heat until the cheese has melted.
Combine the cheese sauce with the cooked vegetables, transfer to a suitable baking dish (mine measured 8 x 8 inches), and set aside to cool for 10 minutes or so. Turn the oven to 190°C (Gas Mark 5 / 375°F) to begin heating up.
Cut a piece of ready-rolled puff pastry to fit the top of your baking dish (you can roll it out a little thinner if needed), with at least an inch excess all the way round. When the vegetable mixture has cooled a little, place the pastry on top, draping it over the edges of the baking dish. It's ideal if it's not sitting directly on top of the vegetables (but it doesn't matter too much). Cut a ventilation hole in the centre of the pastry, and crimp tightly around the edges of the baking dish with a fork. Pulling downwards, to ensure it's sealed thoroughly, tear away any excess pastry. You can use the scraps to create a design for the top of your pie, if you like (I made a few simple leaf shapes) - just place them on top. Lightly brush the top of the pie with milk, and sprinkle on a small amount of finely grated parmesan.
Bake for around 30 minutes, or until the pastry is golden brown, crispy, and puffed up. Serve warm.