Preheat the oven to 425°F. Line a baking sheet with parchment paper.

On a lightly floured board, roll the puff pastry out to remove the fold marks. Cut out 20 squares with a 2-inch cookie cutter. Place the squares on the baking sheet and brush with the beaten egg.

Cover the squares with another piece of parchment paper, and place another empty baking sheet on top. This will keep the dough from rising and cook the squares evenly. Place in the oven and bake for 20 minutes, or until golden brown.

Remove from the oven, remove the top baking sheet and set the pastry squares aside. When cool enough to handle, place on a large serving platter.

In a large sauté pan over medium heat, add the butter and olive oil. Add the onion, thyme and anchovy paste. Stir with a wooden spoon to combine everything, and sauté until brown and caramelized, about 8 to 10 minutes. Remove from heat and cool.

When the onion mixture is completely cooled, spoon about 1 teaspoon of the onion mixture on top of each puff pastry square, covering the entire surface. Top with some shaved Parmesan. Garnish with one of the Niçoise olive halves.