Move the oven rack to the middle position. Preheat oven to 425°F (220°C). Grease one standard 12-count muffin pan with nonstick spray or line with muffin papers. Set aside.
In a large bowl, whisk together the 3 ½ cups all-purpose flour, 3 teaspoons baking powder, 1 teaspoon baking soda, and ½ teaspoon kosher salt. Toss the 1 cup peeled and diced apples and 1 cup fresh or frozen raspberries in the dry ingredients until they are well coated. This will prevent the fruit from sinking to the bottom of the muffins. Set aside.
In a medium bowl whisk together the ½ cup granulated white sugar, ½ cup brown sugar, ½ cup milk, ⅔ cup vegetable oil, 2 large eggs, 1 egg yolk, and 1 teaspoon vanilla extract.
Pour the wet ingredients into the dry ingredients and fold together gently with a spatula until just combined. Be careful to not crush the raspberries and do not over mix. The batter will be thick. Allow the batter to rest for 5 minutes.
Divide the batter between each muffin cup, filling all the way to the top. Sprinkle with coarse sugar.
Bake the muffins at 425°F (220°C) for 5 minutes, then, keeping the muffins in the oven, reduce the oven temperature to 350°F (180°C) and continue to bake for 20 minutes until the tops are golden brown. Use a toothpick to test for doneness. Insert a toothpick into the center of a muffin, if it comes out clean, the muffins are done.
Allow the muffins to cool for several minutes in the muffin pan before transferring them to a wire rack to cool completely.