To make the dough, dissolve yeast in warm water in the bowl of an electric mixer fitted with the dough hook; let stand until bubbly, about 3 minutes. Add milk, melted butter, sugar, eggs, salt, and 3 1/2 cups of flour. Mix on medium speed 3-4 minutes, or until the ingredients are well combined.
Add 1 1/2 cups additional flour and knead on medium speed until dough is smooth and elastic, about 5 minutes. If dough sticks to the sides of bowl, add additional flour until dough doesn’t stick. Form dough into a smooth ball and place into a large greased bowl, turning to coat. Cover with plastic wrap and refrigerate 8 hours or overnight.
To make the glaze, combine diced apple, butter, brown sugar and extract in a saucepan. Cook until butter melts and brown sugar is dissolved. Cook 3 minutes more, or until the apple pieces are semi-soft. Let cool completely before using.
Remove dough from refrigerator and gently punch down. Roll dough on a lightly floured surface to about 1/2-inch thick. Cut dough into 40-45 squares. Gently roll each square into a ball.
To make the sugar coating, combine sugar, cinnamon and apple pie spice in a bowl. Place melted butter in a separate bowl nearby. Grease a 10-12 cup bundt pan with cooking spray.
Dip each ball in melted butter and roll in cinnamon-sugar mixture. Place half coated dough balls in bundt pan and pour half the caramel apple glaze over the top. Place remaining dough into pan and top with the remaining caramel apple glaze. Cover pan loosely with plastic wrap and let rise in a warm place 40-45 minutes, or until doubled in size.
Preheat oven to 350°F. Bake 30-35 minutes or until golden brown on top. Cool 5 minutes before turning out onto a serving platter. Monkey bread is best served the same day it is made, but will keep for 2 days in an air-tight cake carrier.