The best way to thicken Brunswick stew is by adding a flour-based thickener, such as a roux.  To make a roux, melt equal parts of butter and all-purpose flour in a pan over medium heat until it forms a paste.  Then slowly whisk the mixture into the stew while stirring constantly. As it cooks, the starch from the flour will help thicken the stew to your desired consistency.  Another option would be to add mashed potatoes or pureed vegetables such as okra or corn, which are common ingredients for making Brunswick stew thicker and creamier.