In a large mixing bowl, whisk together the all-purpose flour, granulated sugar, baking powder, baking soda, and kosher salt until thoroughly combined.
In a separate medium bowl, whisk the Greek yogurt, whole milk, room-temperature eggs, and vanilla extract until the mixture is smooth and uniform.
Pour the wet yogurt mixture and the 4 tablespoons of melted butter into the dry ingredients. Gently fold everything together with a spatula until just combined; the batter will be thick and may have a few small lumps.
Heat a large non-stick skillet or griddle over medium heat and add a small pat of butter. For each pancake, scoop a heaping 1/4 cup of batter onto the hot surface. Cook for 2 to 3 minutes, until bubbles form on the surface and the edges look set.
Carefully flip the pancakes with a spatula and cook for another 1 to 2 minutes on the second side, until golden brown and cooked through. Repeat with the remaining batter, adding more butter to the skillet as needed.