Preheat the oven to 375 degrees with a rack in the center position.
Melt 1 tablespoon of the butter in a 9- or 10-inch cast-iron skillet over medium-low heat. Add the onion and cook, stirring, until it softens, about 5 minutes.
Add the mushrooms, garlic, 3/4 teaspoon of the salt, the thyme and a generous amount of pepper; cook, stirring often, for about 15 minutes or until the mushrooms' liquid has been released and has evaporated in the skillet and the mushrooms start to turn golden around the edges. (It will seem like a lot of mushrooms at first, but they will cook down.) Remove from the heat.
Meanwhile, combine the eggs, milk, flour and the remaining 1/4 teaspoon of salt in a blender; purée to form a smooth, aerated batter. (If you don't have a blender, whisk the ingredients together in a medium bowl. It's fine if there are a few lumps.)
Transfer the mushroom-onion mixture to a bowl. Rinse and dry the skillet, then return it to the stovetop over low heat. Add the remaining tablespoon of butter; once it has melted and begun to foam, immediately remove from the heat and swirl to coat the skillet. (Be sure the edges of the skillet are coated; this will ease removal after the popover is baked.) Add the mushroom-onion mixture, spreading it into a fairly even layer, then pour in the batter.
Transfer to the oven and bake for 25 to 30 minutes or until the batter is dry on top and feels solid when touched lightly. The popover will be slightly puffed (it deflates fairly quickly), and its edges will have shrunk from the sides of the skillet and become quite brown.