Prep mushrooms and parsley - Destem the mushrooms and brush off any soil with a paper towel. Give the parsley leaves a rough chop and set aside.
Prepare the glaze - Melt 1 T butter in the microwave, then whisk the melted butter together with 1 T oil, 3 T balsamic vinegar, 2 T tamari, ½ t thyme, 1 t brown sugar, and ¼ t each of salt and pepper. Grate the garlic cloves into the mixture and stir to combine.
Poach the mushroom - Add the mushroom caps to a wide pan with ½ cup of water. Cover, and cook on medium-high heat until the mushroom have expelled their water. They'll shrink substantially, and they'll begin to look a bit glossy. Drain the excess water and return the mushrooms to the pan.
Glaze - Pour the glaze over the mushrooms and stir to coat. Continue cooking, uncovered, until the glaze is thick and the edges of the mushrooms begin to caramelize.
Garnish and serve - Add to a serving bowl, sprinkle with chopped parsley, and serve warm.