Warm oil in a heavy skillet or saucepan. Add the garlic, mushrooms, and a dash of salt, cover, and cook on medium-low heat, stirring occasionally, until the mushrooms release their juices and become tender, about 5 minutes.
Add the snap peas, cover, and cook for another 2 to 3 minutes, until they are bright green and just tender. The liquid from the mushrooms should be enough to cook the peas, but add a tablespoon of water if necessary.
Stir in the mint, dill, balsamic vinegar, and salt to taste and serve immediately.