Preheat the oven to 400°F and line a small baking sheet with parchment paper. Peel the apples and cut them into bite-sized pieces.
Melt the vegan butter in a medium-sized pot over medium heat. Add the apples, brown sugar, cornstarch, vanilla, cinnamon, lemon juice, and salt. Stir regularly for 7-8 minutes or until the apples have started to soften.
Let the apples cool down while you prepare the puff pastry sheet. I like to spread them out onto a plate here.
Slightly thaw the puff pastry sheet in the microwave for 15 seconds if it's frozen. Roll it out to a large square, between 10 and 12 inches. Slice it into 4 smaller squares.
Place 1/3 cup of the apple filling into the center of each pastry square. Brush the edges with plant milk. Fold the dough over so that the corners line up to create triangles or rectangles. Press the edges down and then use the tines of a fork to further seal the edges.
Place the turnovers 1 inch apart on the baking sheet. Cut 2-3 small slits into each one and brush the tops with plant milk. Add an optional dusting of ground cinnamon and/or sugar on top.
Bake them for 20-25 minutes, or until they are golden brown.
Prepare the icing while the turnovers are cooling down by combining the powdered sugar and 1.5 tbsp of plant milk. Stir until the icing is smooth. Drizzle it over the turnovers and enjoy!