Boil heavily salted water in a large pot.
Heat the butter in a large pan over medium heat. Once it melts, sauté the diced fennel until it just begins to color, about 3 minutes.
Add the pasta to the salted water and cook for 2 minutes (or a few minutes longer if using store-bought pasta—you want it just short of al dente). Reserve ¼ cup of the pasta water. Add the noodles and reserved pasta water to the fennel pan and cook another 2 minutes on high heat.
Stir in the crab, lemon zest, crème fraîche, and chives. Remove the pan from the heat and season it with the salt, pepper, and lemon juice.
FOR THE FRESH NOODLES: Put the two flours and the salt in the bowl of a stand mixer (or any large bowl if you’re making the pasta by hand), then separately mix together the egg yolks, whole egg, olive oil, and ¼ cup of warm water in another bowl.
Using the dough hook on low speed, combine the dry ingredients (or by hand, if you’re not using a mixer). Slowly add in the egg mixture. Add more warm water as needed, up to another ¼ cup, until the dough begins to form (kneading, if by hand).
Transfer the dough from the bowl to a well-floured surface and, using your palms, fold the dough over itself, repeating until a tight ball forms, about 6 to 8 minutes.
Wrap the dough in plastic and let it rest at least 30 minutes before rolling.
Roll out the dough as thin as possible in a pasta machine, likely in 6 to 8 sheets, dusting with flour as necessary to prevent sticking. Then cut the noodles to a little over ¼ inch wide. Hang them to dry on a chair back or laundry rack as you finish cutting.