In a large Dutch oven or soup pot over medium-high heat, cook the chopped beef until crispy. Remove the beef with a slotted spoon and set aside, reserving approximately 2 tablespoons of beef grease in the pot. Add the butter to the grease.
Add the diced onion, minced garlic, and diced jalapeños to the pot. Sauté for 3 to 5 minutes until the vegetables are softened and aromatic. (Note: For a spicier soup, include some of the jalapeño seeds).
Sprinkle the flour over the sautéed vegetables and stir constantly for about 1 to 2 minutes to create a roux, cooking out the raw flour taste until it smells slightly nutty.
Gradually whisk in the chicken broth, stirring constantly to prevent lumps. Add the diced Yukon Gold potatoes.
Bring the mixture to a boil, then reduce the heat to low. Cover and simmer for 10 to 15 minutes, or until the potatoes are fork-tender.
Turn the heat down to the lowest setting. Stir in the softened cream cheese and heavy cream. Whisk or stir continuously until the cream cheese is fully melted and incorporated into the broth (tempering the cream cheese with a little hot broth first helps prevent curdling).
Stir in the shredded sharp cheddar cheese and the cooked chicken (if using) until the cheese is melted and the soup is smooth and creamy. Season with salt and pepper to taste.
Ladle into bowls and serve immediately, topped with the reserved crispy beef, green onions, and extra jalapeño slices.