To make the sauce, you will need to make a bain marie with a bowl that can comfortably fit on top of a saucepan.
Rub the crushed garlic around the bowl, then add the creme fraiche, anchovies, egg yolks and cheese. Fill the saucepan with a few inches of water and sit the bowl on top, making sure the water doesn’t touch the bottom of the bowl. Bring the water to a simmer and stir the ingredients for 15-20 minutes, until the sauce thickens – it will loosen first and then become thicker, but it does need a little patience. Taste and season with a salt and lots of freshly ground black pepper, and keep warm.
While you are making the sauce, prep the broccoli. Trim the ends and peel any of the tougher stalks, then slice the rest into slender strips to mimic the shape of the tagliatelle.
Bring a large pan of salted water to a boil and cook the pasta according to the packet instructions, adding the broccoli for the last three minutes until it’s completely tender. Drain both, keeping back half a cup of the pasta water, then tip both pasta and broccoli into the sauce bowl, and stir well so that every strand is covered. Wait for a minute and then add a splash of the pasta water if necessary, to loosen things up. Check for seasoning again and serve on hot plates with a little extra cheese.