Preheat your oven to 175°C (350°F). Grease a 9x5 inch loaf pan and set aside.
In a saucepan, melt butter over medium heat. Add diced apples and brown sugar, cooking until the apples are soft and fragrant, about 8 minutes. Remove from heat and let cool slightly.
In a large bowl, whisk together the flour, baking soda, cinnamon, and salt until evenly combined.
In a separate bowl, beat the eggs with sugar until well combined and slightly frothy, about 2 minutes.
Add milk and melted butter to the eggs, mixing until smooth. Then fold in the cooled apples and their juices.
Gradually fold the wet mixture into the dry ingredients with a spatula, just until combined. Do not overmix; the batter should be slightly lumpy.
Pour the batter into the prepared loaf pan and smooth the top with a spatula. Sprinkle generously with cinnamon sugar.
Bake in the oven for 50-60 minutes, until the top is golden and a toothpick inserted into the center comes out clean.
Remove the bread from the oven and let it cool in the pan for 10 minutes. Then transfer to a wire rack to cool completely.
Slice the bread once cooled, revealing a moist crumb with flecks of cinnamon and tender apple pieces. Enjoy warm or at room temperature, perhaps with butter or honey.