01 - Preheat oven to 350°F. Grease and line a 9-inch round cake pan with parchment paper.
02 - Melt 2 tablespoons butter in a skillet over medium heat. Add diced apples, 2 tablespoons brown sugar, and cinnamon; cook for 5 to 7 minutes until apples are tender and caramelized. Remove from heat and let cool.
03 - In a medium bowl, whisk together flour, baking powder, baking soda, salt, and nutmeg until evenly mixed.
04 - In a large bowl, beat 1/2 cup softened butter with 3/4 cup brown sugar until light and fluffy using an electric mixer or whisk.
05 - Add eggs one at a time, mixing well after each addition, then stir in vanilla extract.
06 - Add half of the dry ingredients to the butter mixture, followed by the milk and yogurt. Stir in remaining dry ingredients and mix just until combined. Fold in the cooled caramelized apples.
07 - Pour batter into prepared pan and smooth the surface. Bake for 40 to 45 minutes until golden and a toothpick inserted in the center comes out clean. Allow cake to cool in pan for 10 minutes, then transfer to a wire rack to cool completely.
08 - Melt 1/2 cup butter in a saucepan over medium heat. Stir in 1 cup brown sugar until dissolved. Pour in cream, bring to a boil, then reduce heat and simmer for 3 to 4 minutes until thickened. Remove from heat and stir in salt and vanilla extract.
09 - Serve the cake warm or at room temperature, generously drizzled with warm toffee sauce.