Trim the chard stems, then cut into 2 to 3inch lengths, on the diagonal.
Drop them into the simmering water, add 1/2 teaspoon salt, and cook until tender but still just a little resistant to the knife, about 12 minutes. Scoop them out, reserving the cooking liquid.
Heat the oil in a 10inch skillet with the onion, basil, and saffron threads. Cook over medium heat, stirring occasionally, until the onions soften and the saffron begins to yield its color, about 5 minutes. Add the blanched chard stems, season with salt and pepper, then add a cup of the reserved cooking water. Simmer, covered, until the stems are fully tender, about 7 minutes. Remove the lid, add the tomatoes, and cook until the remaining liquid has reduce to a syrupy consistency.
Transfer the cooked chard stems to a small gratin dish. Scatter the cheese over the top then slide under the broiler until the cheese is lightly browned in places. Garnish with a few more torn basil leaves, then serve