Heat a large saucepan over medium heat and melt the butter in it. Once the butter is melted, add the diced onions and sauté them until they become translucent, but not browned.
In the saucepan with the sautéed onions, sprinkle the flour and stir gently to incorporate it into the mixture. Cook the mixture for about a minute.
Next, add the chicken stock to the saucepan and bring it to a simmer.
In a small bowl, combine the cornstarch and cold water, stirring until the cornstarch is completely dissolved.
Add the seasonings salt, basil, pepper, thyme, dissolved cornstarch water, and instant mashed potato flakes to the saucepan, gently whisking the mixture together over medium-high heat until everything is fully combined.
Whisk the mixture often as you bring it to a boil, then reduce the heat and let it simmer for 5 minutes.
Halve the baked potatoes lengthwise, and then make a few slits in each half to create small, bite-sized pieces. Use a large spoon to scoop out the contents of each potato.
Once you've added the potato pieces, half & half, and 1/3 cup of shredded cheddar cheese to the soup base, bring the mixture to a boil. then reduce the heat and let it simmer for 10 to 15 minutes, stirring occasionally.