Veggie Fried Rice With Frizzled Eggs

Name: Veggie Fried Rice With Frizzled Eggs
Ingredients: 5 tablespoons toasted sesame oil, divided; 8 ounces fresh shiitake mushrooms, stemmed and thinly sliced; 1 teaspoon kosher salt, divided; ½ teaspoon freshly ground black pepper, divided; 4 scallions, thinly sliced, white and green parts separated; 2 teaspoons grated fresh ginger; 2 cloves garlic; 3 cups cold cooked white rice; 2 cups sugar snap peas, halved crosswise; ½ cup frozen sweet peas, thawed; 1 tablespoon low-sodium soy sauce or tamari; 1 tablespoon unseasoned rice vinegar; 4 large eggs