Place a large pot of water on to boil for egg noodles and broccolini. Salt the water and cook the broccolini until tender. Remove to a plate. When the broccolini is cool enough to handle chop it into bite-size pieces.
Place the chicken in a few inches of water with a peeled quartered onion and a couple of bay leaves. Bring the water to a low boil then reduce the heat to a simmer and gently poach the chicken 10 to 12 minutes.
Meanwhile, heat the extra-virgin olive oil in skillet over medium heat and saute the onions and garlic until tender, 8 to 10 minutes. Add the broccolini to the pan and season with salt and pepper. Turn the heat off pan.
While the onions cook, heat the butter in a medium saucepot over medium heat, whisk in flour 1 minute then whisk in the milk and season with salt, pepper, freshly grated nutmeg, and thyme. Thicken 5 minutes then whisk in a little of the sauce to the egg yolks to combine, then add the egg yolks back to the sauce and whisk to combine, turn heat down to the lowest setting.
Cook's Note: If sauce becomes too thick, add a splash of chicken poaching liquid and turn off the heat.
Shred the poached chicken with forks or dice poached chicken into small, bite-size pieces.
Preheat the broiler.
To the large pot of boiling water, add the egg noodles. Cook the noodles according to package directions for al dente. Drain the noodles very well and add them back to the hot pot. Stir in the broccolini, chicken, and sauce, and then transfer to a flameproof casserole dish. Place the casserole in the middle of the oven and broil 3 to 5 minutes until bubbly and brown. Cool and store for make-ahead meal.
To serve the casserole, heat the oven to 375 degrees F. Mix together the panko, Gruyere, Parm, and parsley. Place the casserole on a baking sheet in upper half of the oven and bake 30 minutes, add the topping and bake 15 to 20 minutes more, or until brown and crispy on top.