Melt 1/4 of the butter, and 1/2 of the olive oil in a deep, large nonstick frying pan over medium-high heat.
Sprinkle chicken with salt and pepper to season. Add chicken to hot pan and cook until golden and just cooked through, about 4 minutes per side. Transfer chicken to a plate to cool slightly. Then shred and place into a large bowl.
In the meantime, add 1/4 of the butter and 1/2 of the olive oil to same pan. Add mushrooms over medium-high heat until the liquid from mushrooms evaporates and mushrooms are golden, about 8 minutes. Add onion, garlic and thyme, and sauté until onion is translucent, about 3-5 minutes. Add wine, if using, and simmer until it evaporates, about 2 minutes. Transfer mushroom mixture to bowl with chicken.
Melt remaining butter in the same pan over medium-low heat. Add flour and whisk for 1 minute. Whisk in milk, cream, broth, nutmeg and clove. Increase the heat to medium and simmer, uncovered, until the sauce thickens, whisking often, about 10 minutes. Remove and reserve 1/4 of sauce/mixture and keep warm for serving. Remove pot from heat and whisk in 2/5 of the Parmesan cheese. Test for salt and adjust as needed before adding eggs. Whisk in eggs one at a time, whisking vigorously until incorporated. Set aside.
Preheat oven to 375°F (175°C).
Cook pasta according to package directions. Drain and add spaghetti, sauce, peas and parsley to the bowl with chicken. Toss until sauce coats the pasta and mixture is well blended.
Line a baking sheet with parchment paper and dust 12 areas on the parchment using 1/5 of the Parmesan cheese, spacing them out evenly on the baking sheet.
In a small bowl combine the remaining Parmesan cheese and the panko together and set aside.
Using a fork, twirl pasta into "nests", 1/3 cup at a time and transfer onto parchment paper on baking sheet, placing over the Parmesan cheese. Top with any remaining peas, chicken pieces and mushroom from the bowl and then top each nest with the panko-Parmesan cheese mix.
Bake for 16 to 20 minutes, or until cheese is melted and pasta is lightly browned. Remove from oven and let nests set for 5 minutes.
To serve, spoon 1-2 tsp reserved cheese sauce on a small serving plate, place nest over top. Garnish with chopped parsley and serve.