Heat the oven to 350°F. Grease a 9-by-5-inch metal loaf pan and line it with a parchment sling.
Whisk flour, 1 teaspoon cinnamon, baking powder, baking soda and salt until evenly combined.
Whisk eggs, brown sugar, yogurt, oil and vanilla until smooth.
Fold dry ingredients into wet just until a few flour streaks remain, then fold in the diced apples without overmixing.
Spread batter in the pan. Mix granulated sugar with remaining cinnamon, sprinkle on top, and bake 55 to 68 minutes, tenting loosely if the top browns early.
A tester inserted in a cake area near the center should show moist crumbs, not batter. Cool 15 minutes in the pan, then at least 15 minutes on a rack before slicing.