Prepare a grill for diffused indirect heat by building the hot coals evenly around the perimeter of the grill. Remove the center ring from the grill grate and insert a medium cast-iron skillet. Invert another larger cast-iron skillet over the top. This acts as a pizza oven for your outdoor grill. (If using a gas grill, heat to medium-high heat.)
For the calzone dough: Add the flour, honey, salt and yeast to a food processor and pulse a few times to combine. Then add 1 3/4 cups room temperature water and the olive oil and pulse again until the dough comes together into a rough ball and rides around the bowl. Transfer the dough to a lightly floured surface and knead a few times until a smooth elastic ball forms. Divide the dough into 3 equal pieces and place each in a quart-sized deli container or gallon-sized resealable plastic bag with a drizzle of olive oil. Allow to sit at room temperature for 1 hour, then refrigerate overnight and up to 5 days. This makes enough for 3 calzones. Save the extra dough in the freezer for another time. It will last up to 2 months.
When ready to use, remove the dough from the refrigerator and take it out of its storage vessel while still cold. Allow the dough to rest at room temperature for 1 hour before grilling.
For the cheese mixture: In the meantime, combine the ricotta, mozzarella, pecorino and herbs in a large bowl. Season with salt and pepper.
To make the calzones: On a lightly floured surface, roll out 2 balls of dough into 6-inch rounds. Spoon half of the cheese mixture onto each round, spreading it evenly but leaving a 1-inch border, then fold each round over in half. Crimp the edges with a fork or your fingers to seal it and place them onto the inverted cast-iron. Cover the grill and bake for 15 minutes. Use a pastry brush to brush the tops of the calzones with extra-virgin olive oil, cover the grill and bake until golden brown and the dough is cooked through, another 5 minutes.
Remove from the grill and let sit for a few minutes before digging in!
(Alternatively, you can place the calzones on a parchment-lined baking sheet and bake in a 375 degrees F oven until golden brown and the dough is cooked through, 20 to 25 minutes. Remove from the oven and let sit for a few minutes before eating.)