Preheat the oven to 350°F. Spray a 9x5 inch loaf pan with nonstick spray, and set aside. Stir the apples, and brown sugar together. Set aside.
Whisk the flour, baking powder, baking soda, and salt together in a large bowl. Set aside.
Using a handheld or stand mixer fitted with a paddle attachment beat the butter and sugar together on high speed until smooth and creamy, about 2 minutes.
Add the eggs and then beat on medium speed until combined.
Scrape down the sides and up the bottom of the bowl as needed.
Beat in the yogurt and vanilla extract on medium speed until combined. The mixture will look chunky and curdled– this is normal and expected.
Pour the dry ingredients into the wet ingredients, turn the mixer on low speed, and with the mixer running, slowly pour in the milk. Beat just until combined and do not overmix. You may need to whisk it all by hand to make sure there are no lumps at the bottom of the bowl. The batter will be sticky and slightly thick.
Pour and spread about 1/4 of the batter into the bottom of the prepared loaf pan.
Drain the apples, if you don't it will make the loaf very wet. Mix in the drained apples to the batter that is still in the bowl. Pour and spread the remaining batter on top, making sure to scrape out every last drop of batter.