FOR THE SALTED CARAMEL:
On medium-high heat, melt the sugar, stirring constantly as to avoid burning the sugar.
The sugar will initially start clumping together in lumps, and then will melt into a beautiful deep amber-brown color. Be careful not to burn the sugar.
After the sugar is completely melted, quickly add the butter. Be careful as the caramel will bubble rapidly as the cold butter hits it.
Keep stirring until the butter is melted completely, and then gradually add the heavy cream, stirring continuously. The caramel will once again bubble rapidly. Let the mixture come to a boil and then remove from heat after a minute.
Mix in the salt. Let the caramel cool to room temperature before topping the muffins. The caramel can be refrigerated for two weeks in an airtight container. Gently heat in the microwave if taking out of the refrigerator before using.
FOR THE CINNAMON CRUMB TOPPING:
In a bowl, combine the brown sugar, the granulated sugar, the cinnamon, and the melted butter. Mix in the flour, until the mixture is thick and crumbly. Set aside.
Preheat oven to 220 C. Prepare a 12 muffin pan by greasing it or lining it with muffin liners.
FOR THE MUFFIN BATTER:
In a medium bowl using a handheld electric beater, beat the butter on high speed until pale and creamy, for about 1 minute. Add the brown sugar and granulated sugar and beat on high until creamed, about 2 full minutes, scraping down the sides of the bowl as needed. Add the eggs, yogurt, and vanilla extract. Beat on medium speed for 1 minute, then turn up to high speed until the mixture is combined and uniform in texture. Scrape down the sides and bottom of the bowl as needed.
In a large bowl, combine together the flour, baking soda, baking powder, cinnamon, and salt. Pour the wet ingredients into the dry ingredients and gently mix with a wooden spoon. Add the milk, gently mixing until fully incorporated. Next fold in the apples gently.
Spoon the muffin batter evenly between all 12 muffin tins. Spoon the crumb topping you made onto the tops.
Bake for 5 minutes at 220 C , then keeping the muffins in the oven, lower the oven temperature to 180 C and bake for 12-15 more minutes or until a toothpick inserted in the center comes out clean.
Let the muffins cool completely before topping with the salted caramel.