Set the oven to 400°F (200°C). Wipe the parasol caps clean by hand without rinsing, and cut off the stems. Mince the garlic (4 cloves) and combine it with the olive oil (1 tbsp/15 ml) in a small bowl.
Cover a baking tray with parchment and grease it with a thin film of oil. Place the caps on the tray, undersides up. Sprinkle with salt and black pepper, and work the garlic oil into the gills.
Roast for 10 to 15 minutes, until the caps have shed their juices and taken on some color.
Pour the liquid off the tray and press down lightly on each cap to drain what remains. Spoon the diced tomato (1 medium) over the caps, then add the pesto (3 tbsp/45 ml) in small spoonfuls and the chopped parsley (1½ tbsp).
Grate the mozzarella (10.5 oz/300 g) and cover each cap with it, stopping at the rim so it stays off the parchment.
Drape the prosciutto (3.5 oz/100 g) over the cheese, followed by the diced pepper. Season lightly with dried oregano.
Return the tray to the oven for another 10 to 15 minutes, until the cheese has melted and browned in spots.
Serve hot, whole or cut into wedges, on their own or on slices of warm bread.