In a small bowl, sprinkle the gelatin on 1 1/2 tablespoons of cold water. Stir and set aside for 10 minutes to allow the gelatin to dissolve.
Meanwhile, in a medium bowl, whisk together 3/4 cup of the cream, the yogurt, vanilla extract, and vanilla bean seeds. Heat the remaining 3/4 cup of cream and the 1/3 cup of sugar in a small saucepan and bring to a simmer over medium heat. Off the heat, add the softened gelatin to the hot cream and stir to dissolve. Pour the hot cream-gelatin mixture into the cold cream-yogurt mixture and stir to combine. Pour into 4 (6 to 8-ounce) ramekins or custard cups and refrigerate uncovered until cold. When the panna cottas are thoroughly chilled, cover with plastic wrap and refrigerate overnight.
Combine the strawberries, balsamic vinegar, 1 tablespoon sugar, and pepper 30 minutes to 1 hour before serving. Set aside at room temperature.
To serve, run a small knife around each dessert in the ramkein and dip the bottom of each ramekin quickly in a bowl of hot tap water. Invert each ramekin onto a dessert plate and surround the panna cotta with strawberries. Dust lightly with freshly grated lemon zest and serve.