Put the basil, parsley, garlic, pine nuts, cheese, and olive oil into a blender. Blend until the paste is fairly smooth, adding more oil or some water if it is too thick. Taste and adjust seasoning with salt and pepper; set aside.
Heat a dry cast iron skillet over medium-high heat. Brush the bread slices with olive oil and fry them on both sides until golden brown. Rub them gently with the garlic clove and keep the toast warm. Now add a little oil and the thyme to the pan and fry the prosciutto slices until they are crisp, about 1 to 2 minutes; set aside.
Meanwhile, fill a large skillet with water, bring it to a boil, and add the vinegar and a large pinch of salt. Reduce the heat until the water is just barely bubbling. Crack each egg into a small bowl and carefully slide the egg into the water. Cook until the eggs are set, about 3 1/2 to 4 minutes. Remove the eggs with a slotted spoon and dab the bottom with paper towels to dry them off.
To assemble, put a slice of toast on a plate and place 2 poached eggs on top. Spoon a tablespoon of pesto over each egg and top with some crisp prosciutto. Using a vegetable peeler, thinly shave a few slices of Parmesan over everything and garnish with some basil leaves.
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