In a large dutch oven over medium heat, cook bacon until crisp, about 3 minutes. Using a slotted spoon, move bacon to paper towel-lined plate. When it cools a bit, crumble it.
Add chopped potato skins to pot and cook until browned and crisp, about 8 minutes. Transfer to plate with the bacon.
Add chopped onion to the pot and cook over medium heat until softened, 5-7 minutes. Stir in the garlic and cook until fragrant, about 30 seconds.
Stir in the flour and cook for another minute, before whisking in the broth and making sure to scrape up any brown bits on the bottom of the pot, until smooth. Stir in the potatoes, mustard powder, and thyme and bring the mixture to a simmer. Then, reduce the heat to medium-low, cover the pot, and continue cooking until the potatoes are nice and tender. This will likely take about 7-8 minutes.
Remove the sprig of thyme and put half the soup into a blender and blend until nice and smooth. Return blended soup to pot and whisk in half-and-half.
Put yogurt in a separate, small bowl and then stir in ½ cup of the hot soup (this will temper it) before whisking mixture into the pot of soup.
Stir in the cheese and sprinkle with salt and pepper to taste.
Serve in bowls, topped with bacon, potato skins, and chives.