Cut the stem ends off the eggplants and cut them in half lengthwise. Use a spoon to scoop out the eggplant flesh, leaving a ½-inch (about 1 cm) shell on all four sides.
Place the eggplants in the baking pan, skin side down, and lightly season with salt and pepper. Finely chop the eggplant flesh and set aside.
In the large skillet, heat 2 tablespoons of the olive oil over low heat. Add the onion and cook for about 10 minutes, stirring occasionally.
Add the anchovies and cook for 2 minutes. (optional)
Add the chopped eggplant and another tablespoon of the olive oil, then season with salt and pepper. Cook for 5 minutes, stirring occasionally.
Stir in the pine nuts and cook for another two minutes. The mixture should now be a nice golden brown. Remove from heat and allow to cool 2 to 3 minutes.
Stir in about 1 cup of the chopped tomatoes, bread crumbs, basil, cheese, and parsley. Taste and adjust salt and pepper as necessary.
Spoon the mixture into the four eggplant shells, dividing evenly. Press and form a dome in each. (If preparing ahead, you can now cover and refrigerate up to 6 hours. Bring back to room temperature before baking.)
Add the remaining tomatoes to the pan around the eggplants and season with salt and pepper. Drizzle the remaining olive oil over the eggplants.
Place on the hot grill (or in the oven) and cook for about 40 minutes, until the eggplant shells are soft and the stuffing is hot. If using the oven, you can turn on the broiler at the end and broil for 5 minutes to brown the top.