Ham and Spinach Hash with Fried Eggs

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2 tablespoons extra-virgin olive oil, 2 turns of the pan

4 tablespoons butter, divided

8 small red potatoes

Salt and pepper

1 teaspoon fresh thyme leaves, chopped

1/4 teaspoon crushed red pepper flakes

1 medium red onion, chopped

1 large ham steak, finely chopped

4 extra-large eggs

2 cups baby spinach leaves, chopped or, 1 box, 10 ounces, defrosted frozen chopped spinach, wrung dry in a kitchen towel

10 leaves fresh basil, chopped or torn, a few rounded spoonfuls of prepared store bought pesto may be substituted

2 plum tomatoes, seeded and chopped

1/2 cup grated Parmigiano-Reggiano, a couple of generous handfuls