Preheat oven to 350°F.
To a 9-inch square pan, add the butter. Place in oven to melt; once melted, swirl pan to coat evenly.
Remove from oven and sprinkle brown sugar evenly over the butter. Arrange apricot halves in a single layer over the butter and sugar, cut side down. Set aside.
In a large bowl, combine the flour, baking powder, baking soda, and salt and stir to combine.
Add the granulated sugar, yogurt, vanilla extract, egg, and 2 tablespoons of the reserved apricot syrup. Beat with an electric mixer on low until just combined, then increase speed to medium and beat for 2 minutes.
Pour batter gently over apricot slices and bake until a toothpick inserted into the center comes out clean, 35-40 minutes.
Let cool 10 minutes, then run a knife around the sides of the cake pan and invert onto a serving plater. Serve warm and enjoy!
