Using a smoke tube or similar, cold smoke the beaten eggs for 3 hours.
Pasta
Add the flour to a food processor and blitz for a few seconds. Add the cold smoked eggs and run the food processor until the dough is formed, it takes about 30 to 40 seconds. Add a little water, as needed, to make the dough workable.
Remove the dough from the food processor and place it on a lightly floured surface. Knead the dough for about 1 minute, then wrap in plastic wrap. Let it sit at room temperature for 15 minutes.
Divide the dough into six equal pieces. Roll each piece of dough in a pasta machine until almost paper thin. (Keep the unused dough tightly wrapped, so it doesn’t dry out.)
With a knife, cut the dough into about 1/2-inch (1 1/2cm) wide strips. Dust with flour and place the cut pasta on a baking sheet lined with parchment paper. Cover the pasta with plastic wrap. Repeat with the remaining dough.
Grilled Chicken
Combine the salt, Italian seasoning, black pepper, garlic powder, and onion powder in a small bowl. Season the chicken and let it sit at room temperature while you prepare the grill.
Set the grill for two-zone cooking (direct heat on one side and indirect heat on the other). On the indirect heat side of the grill, set the temperature to 350 degrees F (175 degrees C).
Oil the grill grates and cook the chicken over the direct heat for about 1 minute per side, to get nice grill marks. Move the chicken to the indirect side and let it cook, covered, until the internal temperature reaches 165 degrees F (75 degrees C), 20 to 25 minutes.
Remove the chicken from the grill to a cutting board and let rest while you prepare the Alfredo sauce.
Alfredo Sauce
In a large skillet over medium-low heat, melt the butter. Add the cream and Parmesan and continue to cook, stirring frequently, until the sauce thickens, which takes about 15 minutes. Reduce the heat to low. Fold in the peas, then season to taste with salt and pepper.
Assembling the Dish
Bring a large pot of salted water to a boil. Cook the pasta until just al dente, 3 to 4 minutes, stirring often. Drain the pasta well and add it to the Alfredo sauce.
Slice the chicken on the diagonal.
Divide the pasta among four plates. Top with the sliced chicken and sprinkle with additional Parmesan cheese.