Preheat oven to 425 °F.
Place a medium skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add the onion and garlic to the pan, and sauté until the veggies are tender, 4-5 minutes.
Add the ham and spinach to the pan, season everything with salt and pepper, and cook to heat the ham and spinach through, 2-3 minutes. Remove the pan from the heat and let cool for a few minutes.
While the filling is cooling, spread the pizza doughs out onto the counter. Cut each in half widthwise and in thirds lengthwise, making 12 squares total (6 from each dough).
Portion a small scoop of the ham and spinach filling (about ¼ cup) onto the corner of each square. Brush the edges of each square with a little of the beaten egg and fold the opposite corner over to close in the filling and make a calzone. Press the edges to seal the filling inside and brush the tops lightly with the remaining egg. Arrange all of the calzones on a baking sheet and bake until golden brown, about 15 minutes.
While the calzones are baking, place a medium saucepan over medium-high heat with the butter. Sprinkle the flour over the melted butter and cook for about a minute. Whisk the chicken stock, cider and mustard into the pan, and season with salt and pepper. Bring up to a bubble and simmer until thickened, 2-3 minutes. Reserve warm.
Serve three calzones to each person with some cider-mustard dipper alongside.