Mix the 1 cup sugar and water together in a small saucepan over medium heat. Bring to a simmer and stir until sugar is dissolved. Add thyme and simmer gently for 1 min. Take off the heat and let steep for 10 min. Remove the thyme sprigs. This can be made ahead of time and refrigerated for a few days. When ready to serve, add the simple syrup to a large pitcher along with the freshly squeezed lemon juice and the 5 cups of water, stir well. Serve over ice, garnished with a sprig of thyme and a slice of lemon.