Step 1: Preheat your oven to 400°F (200°C). Toss the cubed apples with 2 tablespoons of sugar and cinnamon. Spread them on a parchment-lined sheet and roast for 15-20 minutes until tender. Let them cool.
Step 2: Reduce oven temperature to 350°F (175°C). Grease a 9-inch springform pan or cake pan.
Step 3: In a small saucepan, melt the butter over medium heat. Continue cooking, swirling the pan, until the butter turns a deep amber color and smells nutty. Remove from heat and let cool for 10 minutes.
Step 4: In a large bowl, whisk the browned butter and brown sugar together. Add the eggs one at a time, then the vanilla extract.
Step 5: In a separate bowl, sift together the flour, baking soda, and salt. Mix in the minced fresh sage.
Step 6: Gradually add the dry ingredients to the butter mixture, alternating with the buttermilk. Start and end with the dry ingredients.
Step 7: Gently fold in the cooled roasted apples until evenly distributed.
Step 8: Pour the batter into the prepared pan and bake for 45-55 minutes, or until a toothpick inserted into the center comes out clean.
Step 9: Allow the cake to cool in the pan for 15 minutes before removing to a wire rack to cool completely.