Heat the oil over medium-high heat in a large (at least 8-quart), heavy-duty pot or Dutch oven. Add the onion, chile, and 1/2 Tbs. salt. Cook, stirring occasionally, until the vegetables are softened and lightly browned in places, 4 to 6 minutes. Add the thyme and cook, stirring, until fragrant, 30 seconds.
Add the broth and bring to a boil. Reduce to a simmer and cook, stirring, until the vegetables are completely tender, 20 to 30 minutes.
Stir in the shredded chicken, potatoes, and corn. Simmer, stirring occasionally, until all are tender and the flavors meld, 5 to 10 minutes.
Stir in 2 Tbs. of the cream. If you would like the soup to be creamier still, add up to the full 1/2 cup of cream. Season to taste with salt and pepper, and garnish each bowl with some bacon and scallion, if you like.