Preheat oven to 350°F (180°C). Line baking sheet with parchment paper; set aside. Halve rolls horizontally; set tops aside.
Hollow out bottom of rolls, leaving about 1/2‐inch (1 cm) border all around. (Reserve hollowed‐out bread for another use.) Place bread bowls on prepared pan.
Arrange slice of ham in bottom of each bowl. Crack 1 egg into each bread bowl. Season with salt and pepper; top with cheese.
Bake until cheese is melted, egg whites are set and yolks are cooked to desired doneness, approximately 20 to 25 minutes. Add reserved tops to baking sheet during last 5 minutes of baking time. Top with salsa and serve with toasted tops.