Cover the carrots with water in a medium-sized pot and boil until soft. Remove from heat, drain, and set aside.
Slice each chicken breast horizontally, parallel to the cutting board, to make two thinner pieces. Pat the chicken breasts dry and season with salt.
Warm oil in a large pot on medium-high heat. Once shimmering, add the chicken. Cook each side for 5-7 minutes or until golden brown. Slice the chicken to ensure it has cooked through. If not, lower the heat and continue to cook until done.
Remove the chicken from heat, cut it into 1/2-inch cubes, and set aside.
Deglaze the pan used to cook the chicken with 1/4 cup of water.
In a separate large pot, bring salted water to a boil. Cook pasta according to package instructions. Drain and set aside.
To make the sauce, add oil to the same pot used for the chicken. On medium-high heat, sauté the onions with salt until softened.
Stir in the oregano, garlic, and mushrooms.
Deglaze the pan with 1/4 cup of water.
Mix in the carrots and butter.
Pour in the heavy whipping cream, green peas, and chicken and simmer.
Stir in parmesan until melted.
Add cooked pasta and ~2 Tbsp pasta water to provide starch to help the sauce adhere to the pasta.
Add more salt to taste.