Open-Faced Sandwiches with Mushrooms and Fried Eggs

INGREDIENTS

4 tsp

extra-virgin olive oil, divided

1 cup

thinly sliced shallots, divided

1 8 ounce package

presliced cremini mushrooms

2 tbsp

dry white wine

1/2 tsp

freshly ground black pepper, divided

1/4 tsp

kosher salt

8 tsp

refrigerated pesto

slices multigrain bread

2 oz

grated fresh Parmigiano-Reggiano cheese (about 1/2 cup)

large eggs

(1/4-inch-thick) slices beefsteak tomato

3 tbsp

chopped fresh basil