Before you start cooking, prepare all your ingredients: grate the garlic cloves and ginger. Meanwhile, fill a large saucepan with salted water and bring it to a boil. This will be used for cooking the gyozas later.
Heat the vegetable oil over medium heat in a large pot. Add the grated garlic and ginger (from Step 1), then sauté for about 30 seconds until fragrant. Pour in the chicken broth and coconut milk. Stir in the soy sauce, fish sauce, toasted sesame oil, Thai red curry paste, brown sugar, and lime juice. Continue stirring until the red curry paste dissolves completely. Bring the mixture to a boil, then reduce heat and simmer for 5 minutes. I like to adjust the lime juice and brown sugar at this point to get the perfect balance of tang and sweetness.
While the coconut curry broth is simmering, add the gyozas or potstickers to the saucepan of boiling salted water (from Step 1). Cook them according to the package instructions, usually about 2 to 3 minutes, until they float to the surface and are fully heated through. Drain the cooked gyozas and set them aside.
Divide the cooked gyozas (from Step 3) among serving plates or bowls. Ladle the hot coconut curry broth (from Step 2) over the gyozas. Top with chopped chives and, if desired, add your choice of vegetables for a pop of color and freshness. Enjoy your flavorful coconut curry gyoza soup! For extra flavor, I like to add a handful of steamed greens or thinly sliced bell peppers.