Preheat your oven to 350°F. If you plan to bake the muffins immediately, liberally grease a 12-cup muffin tin with 1 tablespoon of softened butter. Alternatively, omit the tablespoon of softened butter and use paper liners or baking spray. NOTE: Muffin batter may be made ahead and refrigerated up to 24 hours before baking.
In a small bowl, stir 1 teaspoon cinnamon into 1/4 cup granulated sugar. Mix well and set aside. Peel, core, and chop the apple. Toss in 1 teaspoon cinnamon and set aside. To make the brown sugar cinnamon swirl, stir together 1/4 cup brown sugar, 2 teaspoons all-purpose flour, 1/2 teaspoon sea salt, and 1 teaspoon of cinnamon. Set aside.
In a large mixing bowl, combine the room-temperature eggs with 6 tablespoons of melted butter (cooled), milk, and vanilla at medium-low speed until well combined.
Whisk the remaining 1 1/2 cups flour, 1/4 cup brown sugar, 1/2 cup granulated sugar, baking powder, remaining 1 teaspoon ground cinnamon, and remaining 1/2 teaspoon sea salt in a separate bowl. Add flour mixture to wet ingredients and mix on medium-low until JUST incorporated. It is okay to have visible streaks of the flour mixture.
Add the chopped apple along with any juices accumulated in the bowl. Fold in using a spatula until JUST combined. Use a scoop to evenly distribute the batter to the muffin cups.
Sprinkle the brown sugar cinnamon swirl mixture over each portion of batter. Use the back of a knife to swirl the sugar into the batter.
Bake on the center rack for 22 to 24 minutes or until a pick inserted into the center comes away clean. Begin testing for doneness at the 20-minute mark, as ovens vary.
Remove the muffins from the oven and place the tin on a wire cooling grid for 2 minutes. Use the back of a knife to loosen the muffins if necessary. Spill the tin of muffins onto the cooling rack. Let sit for 10 minutes or until able to handle safely.
Melt the remaining 2 tablespoons of butter while the muffins cool. Use a pastry brush to brush the butter onto each muffin top. Finish by sprinkling the cinnamon sugar over the muffin tops.