1 tablespoon olive oil, divided
1 medium shallot, thinly sliced crosswise
1/2 bunch green asparagus (about 8 medium stalks)
1/8 teaspoon kosher salt
freshly ground pepper
2 cups arugula
1 large beefsteak tomato, cut in 1/4-inch cubes
1/2 ounce feta, crumbled
juice of 1 lemon
1 tablespoon parsley, coarsely chopped
2 large eggs for women; 3 large eggs for men