In a large stock pot on medium-high heat, cool the bacon and olive oil until the bacon is crisp, about 5 minutes. Remove the bacon with a slotted spoon and reserve. Reduce the heat to medium and add the onions and butter to the bacon fat. Cook for 10 minutes until the onions are translucent. Add the chopped garlic and cook, stirring a bit, for another 2-3 minutes.
Stir in the flour, salt, pepper, turmeric and pimento d'esplette if you are using it, and cook for 3 minutes.
Add the chicken stock, potatoes, bell pepper and rosemary. Bring to a boil. Reduce the heat to a simmer and let cook, uncovered for 15 minutes or so until the potatoes are tender.
Add the corn to the soup, then add the half and half, the basil and the cheddar. Cook for 5 more minutes, until the cheese is melted. Season to taste with salt and pepper.
Season to taste with salt and pepper. Garnish with the reserved bacon, a little more fresh basil and extra cheese if you like.