Preheat the oven to 375 degrees F. Bring a large saucepan of salted water to the boil and drop in the chopped broccoli and cauliflower. Boil for 5 to 6 minutes, until cooked, then drain in a colander, reserving the cooking water.
Heat a wide saucepan, pour in a couple of good glugs of olive oil and add the garlic. Fry for a few seconds, then add the thyme leaves, anchovies, anchovy oil and chiles and continue frying for a few seconds more before adding the cooked broccoli and cauliflower with around 4 tablespoons of the reserved cooking water. Stir everything together, put a lid on the pan leaving a little gap, and cook slowly for 15 to 20 minutes, stirring regularly - overcooking the vegetables not only intensies their flavor but gives you the texture that you need for this recipe. Remove the lid for the last 5 minutes to let the moisture evaporate, then use a potato masher to crush the vegetables. Take the saucepan off the heat, taste the vegetables and season carefully with salt and pepper. Spread the mixture on a baking sheet to cool. Meanwhile, get yourself another baking dish or roasting pan (the right size for fitting the cannelloni tubes snugly side by side - I test this by actually laying the tubes into the dish, then remove them and put to 1 side) and pour in the tomato sauce with a pinch of salt and a swig of red wine vinegar.
Now, to make a really quick and easy white sauce, mix the creme fraiche with half the Parmesan, a sprinkling of salt and pepper and a little of the reserved cooking water to thin it down.
Spoon your cooled broccoli and cauliflower mixture into a large sandwich bag and cut off the corner. Twist the top of the bag and squeeze it to pipe the filling into the cannelloni tubes. (If you prefer, use a teaspoon to push the mixture into the cannelloni or use a piping bag.) Fill the tubes up - don't be stingy! - and place them in a single layer on top of the sauce. Lay the basil leaves over the cannelloni and spoon your white sauce evenly over the top. Season with black pepper, sprinkle over the remaining Parmesan and tear over the mozzarella. Drizzle with extra-virgin olive oil and bake in the preheated oven for 30 to 40 minutes, or until golden and bubbling on top.
Dress the arugula leaves with a squeeze of lemon juice and about 3 times as much extra- virgin olive oil. Serve the cannelloni with the arugula and some good crusty bread.