In the same pan, lower the heat to medium, add more oil if needed as well as the onion and oregano. Sauté until the onion smells sweet and is slightly golden.
Add the garlic and sauté it, stirring constantly, for 30 seconds-1 minute. Don't let it burn.
Add the butter and flour and mix.
Turn off the heat and add the stock or water little by little, stirring in between. At first, you will form a thick paste, thinning as you add more. Make sure to use the liquid to scrape the golden bits stuck to the pan.
Turn the heat back on to medium, add the cream and parmesan cheese and stir to incorporate.
Let the mixture come to a boil and correct the level of salt and pepper.
Cut the chicken into strips and add it to the sauce, along with the cooked pasta and a bit of pasta water to bring it all together and to adjust the texture as needed.
Serve with fresh oregano leaves.