Vegetable Gyoza

Name: Vegetable Gyoza
Ingredients: 2 small-medium carrots, diced very small; 8 garlic stems, thinly sliced into rounds; 1 cup cabbage, finely chopped; 5 shiitake mushrooms, diced; 2 green onions, chopped; 1 clove of garlic, finely grated; 1 inch piece of ginger, finely grated; soy sauce, to taste; 1 splash sesame oil; vegetable oil; 1 teaspoon cornstarch; 1 packet round dumpling wrappers