This part can be done ahead of time and placed in the freezer for up to a month. In a small saucepan, combine broth, wine, garlic and thyme and reduce to ¼ cup, cool. Place the butter in the mixer and beat on low speed and add the reduction to the butter. It may take some time for the mixture to emulsify, turn up the speed slowly if necessary. When the butter is completely combined spread onto parchment paper, wax paper or plastic wrap. Shape the butter into a log and roll tightly. Freeze until firm. Separate the ground beef into six equal pieces. Shape the pieces into balls and toss the balls back and forth in your hand until sticky. (This can also be done by placing all of the meat into a stand mixer and mixing on medium speed until sticky.) Flatten each of the balls into a four inch disc. Slice the roll of butter into ¼ slices and place a slice on three of the ground beef patties. Place the other patties on top to create three stuffed burgers. Pinch and shape the edges to seal the butter inside. Grill the burgers for about four minutes per side being careful not to disturb them too much, flip once. Let rest five to ten minutes before serving.