Heat the oil in a medium saute pan or skillet (not nonstick) over medium-low heat. Add the garlic and cook until fragrant, about 2 minutes (do not let the garlic brown). Push the garlic off to the side of the pan and add the mushrooms. Increase the heat to medium high and season the mushrooms with salt and pepper to taste. Cook, stirring occasionally, until the mushrooms are cooked through and begin to become golden in spots, 10 to 12 minutes.
Add the spinach, sprinkle with an additional pinch of salt and turn with tongs until wilted. (If you can't fit all the spinach in the pan, cover it for a few minutes, then uncover, stir and add the rest.) Transfer to a serving platter, drizzle with the vinegar and serve warm.