In a large Dutch oven, melt the butter with the vegetable oil over medium heat.
Add the onion and jalapeño and cook for about 5 minutes or until tender, stirring occasionally.
Add the potatoes, corn, chicken stock, salt, and black pepper.
Bring to a boil then reduce the heat to a simmer. Cook for 15 minutes or until the potatoes are tender.
Add a few ladles of the mixture to a blender so it does not go over half full. Cover the top of the jar with a towel. Pulse the mixture on low being really careful that the heat of the mix doesn't cause the mixture to splatter out of the jar.
Pour the blended mixture back into the Dutch oven.
Add the half-and-half to the soup and warm through. Add more half-and-half for a thinner soup or blend more of the mixture for a thicker soup.
Ladle into bowls and garnish with jalapeno slices and cilantro if desired.
