Preheat oven to 350 degrees F (175 degrees C). Grease an 8-inch square cake pan.
Melt butter and chocolate in the top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching.
Beat eggs in a bowl until light and lemon-colored; gradually beat in sugar. Stir chocolate mixture into egg mixture and beat rapidly for 1 minute.
Whisk flour, baking powder, and salt together in a bowl; add to chocolate mixture and stir to combine. Stir walnuts, apples, and vanilla extract into chocolate mixture until batter is combined. Spoon batter into prepared cake pan.
Bake brownies in the preheated oven until set, about 40 minutes.