In a large saucepan, bring all of the jam ingredients to a boil; season with salt and pepper. Lower the heat to medium and mash with a potato masher. Cook until thick and jamlike, about 20 minutes.
Preheat the oven to 275 degrees . Heat a griddle pan or cast-iron skillet over medium heat. In a large baking dish, whisk together the grated cheese, half-and-half and flour. Whisk in the eggs; season with a few grates of nutmeg and a few grinds of pepper. Soak the bread, 3 pieces at a time, in the egg mixture for a minute or two. Remove 1 piece at a time and press the bread into the shredded cheese. Transfer the bread to a plate. Add 1 tbsp. EVOO to the griddle or skillet and add 3 pieces of bread. Cook, turning, until golden, 5 to 6 minutes. Repeat with the remaining EVOO and bread.
Spread the savory toast with butter, then the tomato jam. If desired, top with an egg and serve with spinach or asparagus.