Preheat oven to 350 degrees and line a baking sheet with foil (if desired, you can cook directly on the baking sheet but foil makes for extra easy cleanup).
For the chicken, set up two wide shallow dishes for the breading process. In one dish, beat egg and add milk, salt and pepper and whisk to combine. In the second dish, add breadcrumbs. Pound chicken breasts to an even thickness approximately 1/2" thick. Dip each piece in the egg mixture, coating both sides and letting any excess drip off before transferring to the breadcrumbs. Coat both sides, lightly pressing the breadcrumbs into the chicken to ensure they are well-coated. Heat a large skillet over medium high heat, add olive oil. When the oil is hot, add chicken breasts and cook about 2-3 minutes per side until golden brown. Transfer to baking sheet and continue cooking in the oven until cooked through, about 5-10 minutes depending on the thickness of the chicken.
For the pasta, heat a large pot of water to boiling over high heat. Generously add salt and add pasta to boiling water. Cook pasta to al dente according to package directions. Drain and toss with pesto sauce.
For the pesto, combine basil, garlic and pine nuts in a food processor and pulse until coarsely chopped, slowly stream in olive oil with processor running until uniform. Stir in cheese and pour over pasta