Combine shrimp, cilantro, garlic, chili paste, brown sugar, salt, fish sauce, turmeric, and cayenne pepper in the bowl of a food processor; pulse until mixture begins to break down. Scrape down sides of the bowl. Continue processing until mixture is very smooth but not puréed and moves around the bowl as a mass; transfer to a mixing bowl. Cover with plastic wrap; press wrap against surface of shrimp mixture. Refrigerate until chilled, 1 to 2 hours.

Place panko on a plate. Scoop out 8 equal-sized portions of shrimp mixture; roll in panko to coat. Shape into thin patties.

Heat oil in skillet over medium-high heat. Add shrimp cakes, in batches; cook until golden brown and cakes spring back to the touch, about 2 minutes per side. Drain on paper towels. Repeat with remaining cakes.