Heat 2 tablespoons olive oil in a saute pan over medium-high heat. Add the garlic and 2 tablespoons thyme and cook for 1 minute. Add the mushrooms and cook, stirring frequently, until they are browned and soft, about 10 minutes. Remove from the heat and let them cool for about 15 minutes. Put the mushrooms into a food processor and pulse a few times until they are finely chopped. (Alternately, you can chop them on a board with a knife.) Scrape them into a bowl. Crumble in the goat cheese and add the remaining thyme. Pour in the truffle oil, if using, and mix well to combine.
Cut the focaccia into 4 pieces, slice them horizontally, and drizzle with some olive oil. On the bottoms place 1/4 of the mushroom mixture and top with some arugula. Put the tops on and gently press each sandwich to flatten it slightly. Drizzle some more olive oil over the tops.
Place each sandwich on a preheated panini press and grill until the bread is toasted and the cheese begins to melt, about 5 minutes. Alternately, preheat a dry cast iron or non stick pan. Place the sandwich in the pan and weight it down with another pan or a clean brick wrapped in foil. Grill 5 minutes, turn the sandwich over, re-weight it, and grill for another 5 minutes. Serve immediately.