Place the two chicken breasts on a cutting board, pat them dry, then cover with parchment and pound to an even thickness using a meat mallet or heavy pan.
Bring a large pot of water to a boil, add 1 tablespoon salt, and cook the pasta for 1 minute less than package instructions; reserve 2 cups of the pasta cooking water, then drain the pasta.
Heat 2 tablespoons olive oil in a heavy-based skillet over medium-high heat. Season the chicken with 1/2 teaspoon salt and 1/2 teaspoon black pepper, then sear 3 minutes per side until cooked through. Transfer to a plate and keep covered.
Reduce heat to low and add the remaining 1 tablespoon olive oil to the same skillet. Add the minced garlic and cook about 30 seconds until fragrant.
Pour in 1 cup chicken broth and bring to a simmer for 2 minutes. Stir in 1 cup heavy cream, 1/2 cup chopped sundried tomatoes, 1 teaspoon dried oregano, 1 teaspoon dried thyme, 1/2 teaspoon black pepper, and 1/2 teaspoon red pepper flakes; return to a simmer.
Remove the pan from the heat and stir in 3/4 cup grated Parmesan until melted, then return to the heat and simmer 1½ minutes until the sauce thickens slightly.
Add the drained pasta and 1/2 cup of the reserved pasta water to the sauce; toss gently and cook 1 minute, adding more reserved water if the sauce becomes too thick.
Tear most of the basil and stir it into the pasta. Slice the chicken breasts and arrange on top of the pasta. Sprinkle with the remaining basil and extra Parmesan to taste.
