Add yeast, sugar, and water to the bowl of a stand mixer. Let sit for 10 minutes – it should be a bit bubbly.
Add oil, salt, and flour. Use dough hook attachment on medium speed on your mixer until it’s well combined. Let rest for another 10 minutes.
While dough is resting, preheat oven to 450 degrees. Also during this time, heat butter in a skillet over medium heat. Add mushrooms and cook for about 5-7 minutes, until tender.
Pour mushrooms into a mixing bowl. Add ham and shredded cheddar cheese. Set aside for now.
When your dough has finished resting, divide it into 6 balls. Roll out each ball in a rectangular(ish) shape.
Spoon your mushroom ham and cheese filling onto one side of each piece of dough. Carefully fold them up, similar to how you would fold up a burrito. Use a fork to crimp down the edges to secure everything, and cut off any excess dough if there happens to be any.*
Cut a few slits in the top of each pocket with a knife. Place the pockets on a baking sheet sprayed with cooking spray.
In a small bowl, whisk together the egg and milk for the egg wash. Brush the egg wash on each mushroom ham and cheese pocket.
Bake at 450 degrees for 12-15 minutes, or until the outside is golden brown and dough is cooked through. Let cool a little and enjoy now, or place in a gallon size freezer bag to enjoy later.
To reheat from frozen, put on a plate and cover with a slightly damp paper towel. Heat for 1:30 minutes in the microwave, or until center is hot (add another 30 seconds if needed).