Preheat the oven to 350°F and grease and line a standard-size loaf pan with a parchment paper “sling”.
In a small bowl, mix the ½ cup light brown sugar and 1 ½ teaspoons ground cinnamon until combined. Set aside.
In a large bowl, whisk together the 2 large eggs and ½ cup granulated sugar until pale, about 2 minutes, then whisk in the ½ cup unsalted butter, ½ cup whole milk, and 2 teaspoons vanilla extract.
Stir and whisk in the 1 ¾ cups oat flour or whole wheat pastry flour, 1 ½ teaspoons baking powder, and ½ teaspoon fine sea salt. Fold in the 1 ½ cups peeled and chopped apples and ½ cup chopped walnuts.
Spoon about ⅔ of the batter into the prepared pan, then sprinkle over the brown sugar-cinnamon mixture evenly over the top. Use a knife to swirl it into the batter, then spoon the remaining batter over the top.
Place in the oven and bake for about 1 hour, until a tester comes out clean or with a few moist crumbs. Transfer to a wire rack and cool for 30 minutes in the pan, then turn out onto the rack to cool completely before slicing.
Notes
Cut the apples into small pieces. I don't mean minuscule, but cut them into a dice which is small enough that they will all cook evenly without overwhelming the loaf with big chunks of apple.
Cool completely. Fruit breads can be delicate, so make sure to cool the bread thoroughly before slicing. This will prevent it from falling apart, especially if you are using oat flour, which often creates a more delicate crumb.
Try not to over mix. When you add the flour and other dry ingredients, stir just until it is incorporated and you don't see any dry streaks. This helps keep the loaf tender.
Toast the walnuts. This is optional, but a few minutes in the oven at 350 F brings out their natural oils and makes them extra flavorful. Spread the walnuts out on a baking sheet, place in the oven for a few minutes, then let them cool before adding them to the batter.
Don't skip the parchment. Lining the loaf pan with a strip of parchment makes it much easier to lift the bread out once it's cool. This is especially helpful with the more delicate oat flour version.