Heat a pot of water to boil for the pasta. Salt the water then add the pasta and cook to al dente or, with a bite to it.
While pasta cooks, heat a medium heavy bottomed pot over medium heat. Add the extra-virgin olive oil and butter. When the butter melts into the oil, add the thyme and grate the onion directly into the pot with a hand held grater or Microplane. Cook the grated onion in butter and oil 1 to 2 minutes, then add flour and cook together 1 to 2 more minutes. Whisk in stock, then combine with butternut squash until warmed through and smooth. Stir in cream or half-and-half and bring sauce to a bubble. Stir in cheeses in a figure 8 motion and season the completed sauce with salt, nutmeg and pepper. Taste to adjust seasonings.
Drain cooked pasta well and combine with sauce. Serve alongside chicken sammies and top with thyme leaves picked from remaining sprigs (Boo would also have a little extra sprinkle of that cheese, please!)