Toss the diced apples, lemon juice and sugar in a medium bowl until apples are well coated.

Grease a sheet pan with the 1/2 Tbsp butter and spread the apples evenly on top.

Bake at 400°F for 12 minutes. Let apples cool before filling turnovers.

In a small bowl, combine 2 Tbsp of water and cornstarch. Mix until fully combined.

In a medium saucepan, heat remaining ½ cup of water on medium-high heat. Once it begins to boil, add the water/cornstarch mixture and continue to boil for 1 minute, stirring constantly.

Remove mixture from heat and add sugar, lemon juice, cinnamon, and vanilla. Stir until fully mixed. Add cherries and stir until coated well. Place filling mixture in the refrigerator until cool.

Whisk together the powdered sugar and milk until a smooth icing forms.

Lay out one piece of puff pastry dough on the counter. Using a knife or pizza wheel, cut the piece of puff pastry into 4 equally-sized squares. (Each square should measure approximately 4.5” by 4.5”.) Repeat with remaining sheet of puff pastry dough.

Place approximately 1/3 cup of apple or cherry filling in the center of one of the pastry squares, leaving ~3/4"-1” border. Brush the border with the lightly beaten egg. Place another square of pastry on top and gently press edges together to seal. Use a fork to crimp the edges. Repeat with remaining pastry squares.

Place filled pastry squares on a parchment-lined baking pan, and sprinkle tops of squares with coarse sugar. Bake at 375 for 32-36 minutes, or until pastry squares are light golden in color. Remove from oven and allow Turnovers to cool slightly before drizzling tops with Icing.