Wrap one slice of prosciutto on bottom of donut and secure with toothpicks. Turn donut quarter-turn and wrap second slice prosciutto on bottom of donut and secure with toothpicks. Repeat with remaining donuts and prosciutto. Place prosciutto-wrapped donuts on parchment paper.
Break egg and slip into center of each donut. Spoon ¼ teaspoon oil over each yolk. Bake in preheated 350° F oven until egg whites are completely set and yolks begin to thicken but are not hard, approximately 10 to 12 minutes. Remove toothpicks before serving.
To ensure food safety, eggs should be cooked until both the yolk and the white are firm. Consuming raw or undercooked eggs may increase your risk of foodborne illness, especially for those with certain medical conditions. For recipes that call for eggs that are raw or undercooked when the dish is served, use either pasteurized shell eggs that have been treated to destroy Salmonella, or use pasteurized egg products.