Mix the patty ingredients, and then gently and briefly form four patties of equal size, each about ¾ inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of the patties to prevent them from forming a dome as they cook. Refrigerate the patties until ready to grill.
Prepare the grill for direct cooking over medium-high heat (400° to 500°F).
In a large skillet over medium heat, warm the vegetable oil, sesame oil, and 1 teaspoon of the chili-garlic sauce. Gradually add the spinach by the handful and stir just until it begins to wilt, 2 to 3 minutes. Remove from the heat and add the sesame seeds.
Combine the ketchup with the remaining 2 tablespoons chili-garlic sauce.
Grill the patties over direct medium-high heat, with the lid closed, until cooked to medium doneness, 8 to 10 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, toast the buns, cut side down, over direct heat.
Build each burger on a bun with chili-garlic ketchup, a patty, and sesame spinach. Serve warm.