Heat 2 tablespoons oil in a large heavy bottom skillet on medium-high heat. Add chicken pieces, salt and red pepper flakes. Brown for about 5 minutes, turn over and brown for 4-5 more minutes. Using a slotted spoon, transfer the chicken pieces to a large sauté pan and cover with a lid.
In the same skillet where you cooked the chicken, sauté mushrooms in one layer until well browned, about 10 minutes. Transfer to the pan with the chicken.
Add one more tablespoon of oil, onions, leek and thyme sprigs and a little more salt; sauté (stirring occasionally), until softened and just beginning to brown, about 6 - 8 minutes. Then add garlic and sauté for another 2-3 minutes. Add the wine to deglaze the skillet and cook until the wine is reduced completely. Discard thyme stems; then add to the pan with the chicken and mushrooms, cover and set aside.
Trim and cut the stems from the Greens into 1/2-inch pieces. Stack and roll the leaves, and cut them thinly. Set aside.
Bring a deep pot with water to a boil over high heat. Add salt and the stems; give them a stir with a wooden spoon to immerse into water. Bring to a simmer, reduce the heat to medium, cover and simmer for 5 to 10 minutes, depending on the thickness of the stems. Add the leaves, gently stirring with the wooden spoon to help them wilt into the water. Cook for about 6 to 8 minutes.
Using a slotted spoon or a strainer, get the cooked greens out of the pot and immediately transfer them into a colander placed in a big bowl with ice water. Once they are cool, wrap in a clean kitchen towel and squeeze out as much moisture as you can. Puree in a food mill or in a processor.
To prepare the Mornay Sauce: melt butter in a pan, sift the flour directly over the butter and stir for about 2-3 minutes to make a smooth paste and to cook out the raw flour taste.
Slowly add warm milk. Simmer on a low heat stirring with a wooden spoon or whisking all the time until sauce thickens about 5 to 8 minutes. Add the nutmeg and season to taste with salt and pepper. Remove sauce from the stove, add the cheese, then beat egg yolks and stir into the Mornay sauce. Add the pureed greens and mix until everything is well combined. Cover with the lid and keep warm.
Preheat oven to 375 degrees F. Spray or butter a large and 2 1/2 to 3-inches deep casserole or glass baking dish (preferably a round one).
Cook linguine al dente, in the same water where you cooked the greens, drain and immediately add to the pot with the chicken, mushrooms, onions, leeks and garlic.
Add the Mornay sauce and gently mix everything together and then pour into prepared baking dish. Spread evenly in the baking dish, sprinkle with cheese. (At this point, you might cool the whole dish, then chill it and bake it later or even freeze up to 3-4 days).
To bake: Cover the dish with parchment paper and foil, place on the middle rack and bake for about 45 minutes. Uncover and bake for another 15 minutes or until the top is golden brown.
Remove the Babka from the oven, let it stand for about 10 -15 minutes and then run a paring knife around the edges and invert onto a large plate. Serve hot or warm as an entrée dish with a light, crispy salad.