Beat butter, ¼ cup brown sugar, and 2 tablespoons yogurt using an electric mixer. Add oats, flour, baking powder, and salt. Mix again with electric mixer until just combined. If it becomes too thick, mix with a spoon or your hands.
Spray a 9-inch tart pan well with nonstick spray. Firmly press crust into the tart pan and up the edges. Place on a baking sheet.
Bake for 6 to 8 minutes or until golden brown. Let cool completely.
For filling:
In a small bowl, sprinkle gelatin over 2 tablespoons of cold water. Let stand for 5 minutes.
Meanwhile, in a small saucepan, warm cream slowly over medium heat until steaming (not boiling). Add gelatin mixture and stir until dissolved (approximately one minute).
In a medium bowl, whisk together yogurt, brown sugar and vanilla bean paste. Stir warm cream mixture into yogurt mixture, until well-combined.
Pour filling into cooled crust. Don't overfill the crust; if you have extra filling, discard it or pour it into a small bowl and eat it separately. Refrigerate the tart at least two hours (until set) or up to one day.
Right before serving, top with fresh fruit and crunchy oat topping (optional).