Prepare the balsamic pearls(Recipe#500406). Pour the olive oil into a tall, clear vessel, like a large beer glass, and place it in the freezer to chill to about 38*F, about 30 minutes.
Combine the agar agar and balsamic in a small saucepan and bring to a boil. Stir to completely dissolve the agar agar. Let cool for 5 minutes, to about 125*F. Then transfer the mixture to a large plastic syringe or a squeeze bottle. Carefully dispense frops into the olive oil to form "pearls", allowing them to sink to the bottom. Drain the pearls when ready to use, reserving the olive oil for another use.
Make the strawberry chips:
Preheat the oven to 300*F. Thinly slice the strawberries lengthwise. Line a baking sheet with parchment or a silicone mat, and arrange the slices evenly on the sheet. Sprinkle with the confectioners' sugar. Bake for 2 hours, or until the strawberries are dehydrated and crisp.
To assemble, halve the fresh strawberries lengthwise. Top with the balsamic pearls, Parmesan shavings, basil flowers, and sprouts.
Garnish with the dried strawberry chips. Dot with a little extra virgin olive oil.