About an hour before you want to assemble these puffs, remove the puff pastry from the freezer and its packaging and allow it to thaw until pliable at room temperature.
Steam the broccoli florets according to the directions on the back of the package.
Drain any liquid off the broccoli and give it a rough chop to make the florets a little smaller, remove and discard any large pieces of stems.
In a small bowl, stir together the chopped broccoli, 1 1/2 cups cheddar, olive oil, salt, and pepper, until well combined.
When the puff pastry is thawed, open it onto a lightly flowered surface, and cut each of the two sheets into 6 equal rectangles. You should have 12 equal rectangles in the end.
In a small bowl, whisk together the egg and a splash of water to create an egg wash.
Arrange 6 of the pieces of puff pastry on a parchment-lined sheet pan. Prick them with a fork, 10-12 times. Ribbon one slice of ham onto each piece of puff pastry leaving about 1/4 inch of the pastry showing around the perimeter. Top with the broccoli cheddar mixture.
Brush the edges of the pastry that are still showing with the egg wash, and top each stack with a remaining piece of pastry, stretching it over the mound to encase it. Press the top piece of pastry into the bottom piece of pastry, in the spots that have been brushed with the egg wash. This should create a seal. Use a fork to crimp all the way around the edges to create a good seal.
Using a sharp knife, cut three slits in the top if each pocket you just created to let steam out.
Brush the tops with egg wash and top with the remaining cheddar cheese.
Place the pockets into the refrigerator for 15-20, then preheat the oven. Do not remove the pockets until they are ready to go in the oven.
Preheat the oven to 425˚F.
Bake in the hot oven for 22-24 minutes, until golden brown.
After baking, remove the puffs to the cooling rack over a sheet pan. This will keep the bottom of your pastry crisp and allow any lingering moisture to be released.
Allow time to cool, and enjoy!