Position a rack in the upper third of the oven and preheat to 450. Toss the tomatoes, olive oil and 1/4 teaspoon salt in a large ovenproof skillet. Roast in the oven until the tomatoes are charred around the edges, about 25 minutes. Reduce the oven temperature to 300 but keep the tomatoes inside.
Meanwhile, bring 5 cups water to a boil in a saucepan over medium heat. Slowly whisk in the polenta until smooth and creamy. Add 1 teaspoon salt, and pepper to taste. Remove from the heat, cover and keep warm.
Bring a pot of salted water to a boil. Slice the chard leaves into wide strips and the stems into 1-inch pieces. Boil the stems until almost tender, about 5 minutes, then add the leaves and cook until both are tender, about 3 more minutes. Drain the chard.
Remove the skillet from the oven and place over medium-high heat. Push the tomatoes to one side, add the butter and swirl until the butter is golden brown. Add the chard and toss to coat. Divide the polenta among 4 bowls. Top with the tomatoes and chard. Season the cheese with salt and sprinkle over the top.