In a small saucepan, combine the strawberries and granulated sugar and cook on low for 20 minutes, or until the strawberries begin to break down.
Mash the strawberries with a potato masher and cook for another 20 minutes, until a chunky sauce forms. Remove the strawberry sauce from the heat, allow it to cool, and then cover it with plastic wrap and put it in the fridge to chill for 2 hours. Tip: You can do this step a day ahead.
In a small bowl, whisk together the lime juice and lime curd.
In the bowl of a stand mixer fitted with a whisk attachment, beat the cream and powdered sugar on high until stiff peaks form.
Set the mixer on low and whisk the lime mixture into the whipped cream.
Add the lime zest to cream and, with the mixer set on low, whisk until just incorporated.
Remove the bowl from the stand mixer and fold in the vanilla, Greek yogurt and ¾ of the chilled strawberry sauce.
Line a glass dish measuring 4×8 inches with plastic wrap, leaving an overhang of 4 inches on all sides.
Spoon the semifreddo mixture into the pan and fold the plastic wrap over it. Put the pan in the freezer for at least 3 hours.
Lift the frozen semifreddo onto a cutting board using the overhanging plastic to transfer it. Remove the plastic and cut the semifreddo into 8 squares.
Serve on plates, topped with the remaining strawberries, slices of Key lime and a drizzle or dot of lime curd.