Heat the olive oil in a skillet over medium-low heat. Cook the garlic in the oil until just fragrant, about 1 minute.
Add the Swiss chard to the garlic and cook until wilted, 3 to 5 minutes; transfer the chard and garlic to a bowl, reserving any liquid in the skillet and returning it to the heat.
Cook the mushrooms and onion to the reserved liquid in the skillet until warmed, 2 to 3 minutes.
Pour the balsamic vinegar over the mushroom and onion mixture; cook and stir another 2 to 3 minutes; add to the bowl with the chard mixture along with the tomatoes. Gently mix to incorporate. Toss with the blue cheese. Season with salt and pepper to serve.