Step 1Heat the oven to 225 degrees. Cut the Roma tomatoes in half lengthwise and place them cut-side up on a rack set on a baking sheet. Drizzle 2 tablespoons of olive oil over them. Sprinkle with one-half teaspoon salt and a few grinds of black pepper.
Step 2Roast the tomatoes for 4 to 4 1/2 hours, until they are very tender and begin to collapse and show some browning. Remove from the oven and let cool, then coarsely chop. Add additional salt and pepper to taste if necessary. Stir in the balsamic vinegar.
Step 3Using a mortar and pestle, work the garlic and one-fourth teaspoon salt into a fine paste. Grind in the pine nuts and basil until a smooth paste begins to form. Slowly drizzle in one-fourth cup of olive oil, then work in the Parmigiano cheese, mixing to combine. Just before serving, stir in the lemon juice and adjust seasoning if necessary.
Step 4Add olive oil to a large skillet until the oil fills the pan to about one-fourth inch deep. Heat the oil over medium-high heat and when the oil is hot but not smoking, add the prosciutto slices; they should sizzle lightly. Cook until nearly crisp, 4 to 4 1/2 minutes. Drain on a paper towel and let cool before breaking into 6 shard-like pieces for garnishing the verrines.
Step 5For assembly, spoon about one-third cup roasted tomatoes into each of six glasses, about 3 1/2 inches high by 3 inches in diameter.
Step 6Spoon a generous 2 tablespoons of burrata cheese over tomatoes to make an even layer in each of the glasses. Top with about 1 tablespoon pesto in each glass, spreading it to the sides of the glass. Spoon 1 tablespoon burrata into the center of each glass, leaving an edge of the green pesto showing. Drizzle a little olive oil over the burrata and sprinkle cracked pepper and fleur de sel. Top each glass with a shard of prosciutto.