Kimchi Fried Rice

Name: Kimchi Fried Rice
Ingredients: 4 medium scallions; 1 cup drained Napa cabbage kimchi; 1 tablespoon mayonnaise; 1 tablespoon kimchi brine; 1 tablespoon granulated sugar; 1 tablespoon coarse gochugaru (Korean pepper flakes); 1 tablespoon plus 1 1/2 teaspoons soy sauce; 2 tablespoons unsalted butter; 2 cups cold cooked white rice; 1 tablespoon toasted sesame oil; 1/4 teaspoon kosher salt, plus more as needed; 1 tablespoon neutral oil, such as canola or vegetable; 2 large eggs; Toasted sesame seeds, for garnish