Position a rack in the center, or 1 rung above the center, of the oven and preheat to 450°.
Slice off the tops of the tomatoes and, using a spoon, gently scoop out the center of each tomato; discard the tomato pulp. Season the tomatoes with salt and pepper. Line each tomato with a slice of ham, allowing it to hang over the sides.
In a food processor, pulse or grind the bread into large homemade breadcrumbs.
In a medium skillet, heat the oil, three turns of the pan, over medium to medium-high. Add the onion, garlic, and thyme; season. Cook, stirring occasionally, until the onion softens, about 5 minutes. Stir in the breadcrumbs. Transfer the mixture to a small bowl and mix in the cheese and parsley. Stuff the tomatoes with the breadcrumb mixture. Bake until light golden on top, about 15 minutes.