Heat 1 tablespoon olive oil in a soup pot or dutch oven over medium heat.
Add bacon and cook, turning occasionally, to a desired crispness, about 4 to 5 minutes.
Remove bacon and transfer to a paper towel-lined plate; set aside. Leave bacon drippings in the pan.
Add remaining olive oil; stir in onions and cook for 2 minutes.
Add garlic and cook, stirring, for an additional minute.
Stir in potatoes and add chicken broth; bring to a boil.
Reduce heat to a simmer and cook until the potatoes are tender, about 15 minutes.
Using a slotted spoon, transfer half the potatoes to a bowl and mash until almost smooth.
Return the mashed potatoes to the soup pot.
Add yogurt, sour cream, and 1/4 cup of cheese.
Cook and stir until the cheese is melted and the soup is heated through, about 2 minutes.
Remove from heat and taste for seasonings; adjust accordingly.
Crumble the previously prepared bacon.
Ladle soup into bowls and garnish with bacon crumbles, remaining cheese and green onions.