Make the Hollandaise: Heat the olive oil and butter in a pan until the butter is melted. Put the egg yolks, cream, lemon juice, salt, and pepper into a blender and blend for a few seconds at high speed until you have a smooth frothy mixture. Still at high speed, start pouring in the hot butter mixture in a thin, steady stream, not too slowly. As you add the butter mixture, the sauce should thicken. Continue pouring and blending until all the butter is used. Serve immediately or keep warm.
Make the eggs: Fill a large deep skillet halfway with water and bring to a simmer. Brush 1 tablespoon butter into 2 ramekins or small ovenproof bowls. Sprinkle with salt and pepper. Crack 2 eggs into each dish. Place the dishes into the simmering water and cover the pan. Cook until firm, about 6 to 7 minutes.
While the eggs poach, heat a grill pan and grill the prosciutto until crispy. Brush the pita with the remaining butter and grill.
Put a piece of toasted pita on each of 2 plates and top with the prosciutto. Remove the eggs from the ramekins by running a knife or small spatula around the eggs. Unmold the eggs on top of the prosciutto. Drizzle the eggs with 2 tablespoons Hollandaise, grind a little black pepper over each, and serve immediately.
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