Toast pine nuts and garlic cloves in a dry saucepan over medium heat. Transfer to a food processor; add basil, 2 tablespoons olive oil, Parmesan cheese, salt, and pepper. Process until combined.
Turn on a multi-cooker (such as Instant Pot(R)) and select the Saute function. Coat the bottom of the pot with the remaining olive oil.
Season both sides of chicken with salt and pepper; place into the pot, breast side-down. Cook until browned, 5 to 7 minutes. Flip; continue browning for 5 minutes. Turn off Saute function. Rub some pesto over the chicken. Mix in the remaining pesto, milk, and sun-dried tomatoes.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 25 minutes. Allow 10 to 15 minutes for pressure to build. Let rest, 5 to 10 minutes; release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove lid.
Transfer chicken to a separate dish; separate all meat from the bone. Discard bones and skin. Cut meat into bite-size pieces; place back in with the sauce and stir to combine. Season with salt and pepper.
Bring a large pot of lightly salted water to a boil; add penne and cook, stirring occasionally, until tender yet firm to the bite, about 11 minutes. Drain. Add pasta to the chicken and sauce; stir to combine.