Spray a round cake pan with non-stick spray. Line pan with plastic wrap both vertically and horizontally so all sides are covered, leaving several inches hanging off the sides.
Arrange beet slices to cover the bottom of the pan. Top with thin layer of goat cheese. Repeat until pan is filled with layers to the top. Gently fold plastic wrap over the top both vertically and horizontally, so all is sealed.
Refrigerate overnight. To unmold, unwrap plastic wrap and place face-side down on cutting board. Remove pan, then peel off plastic wrap. Using a biscuit cutter warmed in hot water, cut terrine into 12 rounds. Remove with spatula and top with edamame and parsley. Garnish with edible petals and/or drizzle with balsamic syrup (optional).
Three towers per appetizer serving.