Heat the olive oil and butter in a roasting pan over medium-high heat. Fry the pork cheeks on all sides until well browned. Remove from the pan and set aside. Add the chopped onion to the pan, sprinkle with 1 teaspoon sugar, and cook over medium heat for 5 minutes.
Return the pork cheeks to the pan. Pour in the cider sec, then add the diced potatoes, sliced carrots, 1 teaspoon honey, and the bouquet garni. Season with salt and pepper. Cover and simmer over low heat for 1 hour 30 minutes until the pork reaches an internal temperature of 63°C (145°F).
Add the diced turnips and continue cooking over low heat for 30 minutes. Transfer the pork cheeks and vegetables to a serving dish. Increase the heat to medium-high and cook the remaining liquid, stirring occasionally, until it reduces and becomes syrupy.
Pour the sauce over the pork cheeks and vegetables and serve warm.