Place eggs in a saucepan; cover with water. Bring to a boil, remove from heat, and let eggs stand in hot water for 15 minutes. Remove eggs from hot water, cool under cold running water, and peel.
Cut each egg in half lengthwise; place egg yolks in a bowl. Mash yolks with a fork; stir in dill, mayonnaise, and Dijon mustard.
Place egg whites cut-side-up on a serving platter. Cut each prosciutto slice into 6 pieces; lay one piece into each egg white half. Fill egg whites with yolk mixture; sprinkle with black pepper, top with a cucumber slice, and sprinkle with paprika. Refrigerate, covered, for 30 minutes before serving.
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