Allow frozen puff pastry to thaw to room temperature and spray two baking sheets with cooking spray.
Place each pastry rectangle on a separate baking sheet.
With a sharp knife, draw a large rectangle - about ¼ inch from the edge - around each pastry. Be careful not to cut through the dough. Then, make small marks around the outside edge with the fork.
With a pastry brush, brush some egg wash all around the border. Bake at 400 degrees for 15 minutes.
Chop cherry tomatoes in half, squeezing each tomato half over the sink to release water.
Add all tomatoes to a bowl and drizzle with olive oil, 1 tbsp chopped basil, and the garlic powder.
Chop the white onion and the green onion into small pieces.
Once the puffed pastries come out of the oven, add ½ cup creamy salsa or pesto into each rectangle (avoiding the border). Keep the layer thin. (If the pastries puffed up while baking, you will gently press the puffed areas back down before adding the pesto.)
Cover salsa with the asiago cheese.
Add the seasoned tomatoes (cut side facing up) and sprinkle the chopped white and green onions on top.
Bake the puffed pastries again for 15-20 minutes at 400 degrees, checking to make sure the edges turn a nice golden brown and not too dark.
Remove from the oven and cut each tart into nine squares with a sharp knife or pizza wheel.
Garnish with reserved basil and the parmesan cheese. Serve warm.