For the Streusel Topping

1. In a small bowl, stir together the melted butter, brown sugar, granulated sugar, flour, pecans, and cinnamon. Mix until all the ingredients are moistened and the mixture looks like wet sand.
2. Spread the mixture onto a plate or shallow dish and place it in the refrigerator to chill while you prepare the muffin batter. Chilling helps it firm up into a perfect crumble consistency.

For the Apple Cinnamon Muffins

1. Preheat your oven to 375°F (190°C). Place 12 paper liners in a standard muffin tin.
2. In a small bowl, gently toss the diced apples with the lemon juice. This helps prevent them from browning. Set the bowl aside.
3. In a large bowl, whisk together the sugar, flour, baking powder, cinnamon, and salt.
4. In a separate medium bowl, combine the eggs, applesauce, melted butter, and vanilla extract.
5. Pour the wet ingredients into the bowl of dry ingredients. Mix gently until they are almost combined but a few streaks of flour remain.
6. Carefully fold in the diced apples until the batter is just mixed. Be careful not to overmix.
7. Divide the batter evenly among the 12 muffin liners, filling each one about three-quarters full. You should use all of the batter.
8. Remove the chilled streusel from the refrigerator. Crumble a generous amount over the top of each muffin, breaking it up with your fingers.
9. Bake for 17 to 20 minutes. The muffins are done when a toothpick inserted into the center comes out clean. Let them cool while you make the caramel.

For the Caramel Sauce

1. Pour the granulated sugar into a medium saucepan over medium heat, spreading it into an even layer. Let it melt slowly without touching it. This process takes a few minutes, so watch it closely.
2. Avoid the temptation to stir the melting sugar. If some parts are not melting, you can give the pan a gentle shake to help distribute the heat.
3. Once the sugar is almost fully melted and has turned a nice amber color, warm the heavy cream in the microwave for about 45 to 60 seconds until it is steaming.
4. Remove the saucepan from the heat and place it on a heatproof surface like a towel. Carefully pour the hot cream into the sugar while whisking quickly. Stand back, as the mixture will bubble and steam.
5. After the cream is mixed in, stir in the butter cubes, vanilla, and salt. Finally, whisk in the sifted powdered sugar until the sauce is smooth.
6. Spoon the warm caramel over the cooled muffins. The sauce is easiest to drizzle while it is still warm, as it will thicken as it cools. Enjoy!