1. Finely chop the broccoli and transfer it to a microwave-safe bowl. Add 1 tablespoon water, then cover the bowl and microwave it for 2 minutes (the broccoli should be lightly steamed and very green). Drain and let it cool.
2. Unfold one of the pastry sheets onto a piece of plastic wrap. Dust the pastry and a rolling pin with flour, then roll out the pastry into a 10-inch square.
3. Generously coat the pastrys surface with the egg wash. Scatter the steamed broccoli on top, then layer on the ham slices and uniformly sprinkle on the Cheddar.
4. Starting with the end closer to you, roll the dough one turn toward the center. Repeat with the opposite end. Continue rolling, alternating ends, until the two meet in the middle. Lightly brush egg wash along the center. Tightly wrap the roll in plastic and trim the ends so that theyre even. Chill the roll in the refrigerator for 30 minutes. Heat the oven to 400 degrees .
5. While the broccoli roll chills, make the sun-dried tomato and pesto roll following steps 2 through 4. To layer the filling, spread on the pesto, scatter the sun-dried tomatoes, and sprinkle on the mozzarella.
6. While the pesto roll chills, line a baking sheet with parchment paper. With a serrated knife, unwrap the ham roll and cut it into 16 slices, then transfer the slices to the prepared baking sheet.
7. Bake the palmiers until theyre puffy and lightly browned, about 18 minutes, then slice and bake the second roll. Let the palmiers cool on a wire rack before serving.