Preheat the oven to 350°F (175°C). Grease an 8x4-inch loaf pan with some of the ½ cup oil and lightly dust the pan with flour, tapping out any excess.
In a large bowl, whisk together 1 ½ cups all-purpose flour, 1 cup granulated sugar, 1 teaspoon cinnamon, ½ teaspoon baking soda, ¼ teaspoon baking powder, and ¼ teaspoon salt until evenly combined.
In a separate bowl, whisk the 2 large room-temperature eggs, ½ cup oil, and ½ teaspoon vanilla until smooth.
Pour the wet mixture into the dry ingredients and stir with a spatula or wooden spoon until just combined. Do not overmix; the batter will be thick.
Fold into the batter the 1 cup peeled and chopped apples and half of the 1 cup chopped walnuts until evenly distributed.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
In a small bowl, toss the remaining walnuts with the 1 tablespoon brown sugar, then sprinkle this mixture evenly over the top of the batter.
Bake for 50–55 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs attached.
Let the loaf cool in the pan for 10 minutes, then remove it from the pan and transfer to a wire rack to cool completely before slicing.