Combine the vegetable broth, and 2 ginger slices in a medium saucepan; bring to boil. Add wild and brown rice and bring to boil. Reduce heat to medium-low; cover and simmer until the rice is just tender, 50 to 55 minutes. Drain if necessary. Discard ginger slices. Set rice aside.
Meanwhile, position the oven rack in the center of the oven and preheat to 400 degrees F. Lightly oil a 13-by-9-by-2-inch glass or ceramic baking dish with 1 tablespoon vegetable oil; set aside.
Heat the remaining 2 tablespoons oil in a large nonstick skillet over medium-high heat. Add the onion, carrots, salt, pepper, green onions, garlic, ginger, 1 1/2 teaspoons curry powder, and cumin. Cook, stirring occasionally, until the vegetables are tender, about 6 to 7 minutes. Add the coconut milk and bring to a simmer. Cook for 1 minute until the sauce coats the back of a spoon.
Remove from heat and let cool while preparing the chard leaves.
Bring a large pot of salted water to boil. Using a sharp knife, cut away the thick stems from the center of each leaf; discard the stems. Cut each leaf in half lengthwise. Trim the wide ends from the leaves to make each leaf half about 10 inches long. Using tongs and working with 1 leaf half at a time, dip the leaf half into the pot of boiling salted water for 10 seconds. Transfer to paper towels to drain.
Place about 1/3 cup of the rice filling onto the end of one Swiss chard leaf half. Roll up the leaf, enclosing the filling completely. Repeat with the remaining leaf-halves and rice filling for a total of 12 rolls.
Mix the remaining 1/2 teaspoon curry powder into the tomato sauce. Spread 1 cup of the sauce onto the bottom of the prepared baking dish. Place the filled rolls, seam side down, in single layer atop the sauce in the dish. Spoon the remaining tomato sauce over the rolls. Bake until the rolls are heated through, 25 to 30 minutes. If desired, garnish with additional sliced green onions and serve.