Preheat the oven to 450°F (230°C).
Place the baby Yukon gold potatoes in a large pot and cover with cold water. Add 1 teaspoon of kosher salt to the water.
Bring the water to a boil over high heat and cook the potatoes until tender when pierced with a fork, about 15-20 minutes.
Drain the potatoes and let them cool slightly until they can be handled safely.
Line a large baking sheet with parchment paper or lightly grease it with olive oil.
Place the cooked potatoes on the baking sheet, spacing them evenly.
Using the bottom of a sturdy glass or a potato masher, gently press down on each potato until it is flattened to about 1/2 inch thickness, taking care not to break them apart completely.
Drizzle the smashed potatoes evenly with 1 tablespoon olive oil and sprinkle with 1/2 teaspoon black pepper and additional kosher salt to taste.
Bake the potatoes in the preheated oven for 20 minutes, or until the edges are golden and crispy.
Remove the baking sheet from the oven and sprinkle the shredded sharp cheddar cheese evenly over the potatoes.
Return the baking sheet to the oven and bake for an additional 5-7 minutes, or until the cheese is melted and bubbly.
Remove the potatoes from the oven and let them cool for 2 minutes.
Top each potato with a dollop of sour cream, a sprinkle of sliced green onions, and a dusting of smoked paprika.
Serve immediately while warm and crispy.