Place pesto ingredients in the food processor and pulse until combined. If your processor is parve, you can add the Parmesan cheese by hand after the rest of the ingredients are pureed.

Moisten the edges of a wonton wrapper and place one salmon cube in the center. Top with a half teaspoon of pesto. Bring the corners up to the center and pinch closed. Be sure to pinch all along the edges so the wonton does not open.

Heat oil for deep-frying over medium to medium-high heat until a deep fry thermometer measures 375 degrees Fahrenheit. Drop three to four wontons in the oil. Deep-fry in hot oil on one side. Flip over and fry on the other side until golden brown, three to four minutes per side. Drain on a cooling rack or paper towels.

Whisk together mayonnaise and sweet chili sauce and serve with the wontons.