Pre-heat the oven to 350 degrees, then grease a 9-inch spring-form pan. Place over an aluminum foil lined baking sheet. You'll thank me later for clean up.
In a small sauce pan, combine 1/2 stick of butter and brown sugar over medium high heat. Once boiling and thick, pour into spring-form pan.
Layer the mango slices like a fan over the sauce you just poured it. Let it hang out a bit.
In your stand mixer, toss in butter and regular sugar and beat on high until somewhat fluffy, about 5 minutes. Scrape down the sides when you see it becoming a paste and sticking to the edges.
Meanwhile, mix all your dry ingredients together and set aside.
Add the eggs, one at a time. Once they're just about combined (you can no longer see the yolk streaks), pour in vanilla.
With the mixture on low-we don't need a flour hurricane, add half the flour mixture.
When you no longer see flour, add half the yogurt. Mix well. Then add flour, mix well, then last of yogurt.
Pour the cake mixture over the mangoes. Using a spatula, spread evenly.
Bake for 45 minutes, or until a toothpick comes out clean.
The center of the cake will deflate, it's OK! The rest of the cake will deflate as well and you're going to flip it over, so no worries.
Cool in the spring-form pan on a wire rack for 10 minutes. Then flip onto a serving platter and enjoy!