Preheat the oven to 425°F. Rub the potatoes lightly with olive oil and prick them several times with a fork. Arrange on a lined baking sheet and bake for 45 minutes to 1 hour, turning every 20 minutes, until soft all the way through. Set aside until cool enough to handle, then peel and cut into bite-sized pieces.
Set a large pot over medium heat and melt the butter. Add the onion and cook for about 3 minutes, until softened. Stir in the garlic and cook for another minute.
Sprinkle in the flour, thyme, and basil, stirring to coat the onions. Whisk in the broth slowly until smooth. Bring to a gentle boil, then reduce the heat to low and stir in the half-and-half.
Add the baked potatoes and cayenne pepper sauce, if using. Simmer for about 5 minutes, just until everything is heated through. Adjust seasoning as needed.
Ladle into bowls and finish with your favorite baked potato toppings.