Bring a large pot of salted water to a boil. Add rapini, return to a boil, and cook until stems are crisp-tender, 2 to 3 minutes. Remove from the water with a slotted spoon and allow to drain completely. Save the pot of water to cook the penne in.
Heat olive oil in a large frying pan over medium heat. Add red onion and cook for 2 minutes. Add garlic and red pepper flakes; cook for 30 seconds. Add rapini and white wine; cook until the wine is reduced by half.
Meanwhile, return the pot of water to a boil. Add penne and cook until almost tender, about 10 minutes. Drain penne, reserving 1/2 cup of the starchy cooking water.
Add the penne to the pan with the rapini. Add chicken, reserved cooking water, and chicken bouillon. Continue to cook until penne is tender yet firm to the bite, 2 to 3 minutes. Season with salt and pepper. Stir in 1/4 cup Parmigiano-Reggiano cheese. Divide among serving bowls and serve with remaining cheese.