Set two oven racks in the oven and preheat to 350 degrees F.
Put the cream, rosemary leaves and thyme sprigs in a small saucepan over medium heat. Bring to a simmer, then remove from the heat and season with salt and pepper. Let stand at room temperature while you proceed.
Butter two 4-ounce ramekins with the softened butter. Line each ramekin with two slices of the prosciutto. Set the ramekins on a small baking sheet and bake until the prosciutto is crisp, 10 to 15 minutes.
While the prosciutto crisps, cut each slice of ciabatta into two sticks. Toss with 2 tablespoons of the olive oil and some salt and pepper. Transfer to a baking sheet and bake until lightly golden brown and crisp on the outside but still tender, about 8 minutes. Toss with the chives and set aside.
Meanwhile, heat the remaining 2 tablespoons olive oil in a small nonstick skillet over medium heat. Add the grape tomatoes, sugar and some salt and pepper; cook, tossing occasionally, until lightly caramelized and starting to collapse but still intact, about 5 minutes. Stir in 1 teaspoon of the chopped thyme and set aside.
When the prosciutto is crisp, remove the baking sheet from the oven. Add 1 egg to each cup, and season with salt and pepper. Bake until the whites are set around the edges, about 4 minutes. Spoon some of the cream over the eggs, return to the oven and bake until the whites are completely set, and the yolks are just set but still runny, another 3 to 4 minutes.
Transfer the ramekins to napkin-lined saucers. Top each ramekin with some Parmesan, a spoonful of the tomatoes and a drizzle of olive oil. Serve with the toasted ciabatta.