Place eggs in a large saucepan and cover with cold water. Bring to a boil over high heat, cover pan, remove from heat, and let sit 13 minutes. Drain, tap eggs to crack shells all around, and let eggs sit 10 more minutes in very cold water. Peel and halve eggs.
Meanwhile, pick leaves from watercress until you have 1 cup of lightly packed leaves; set remaining watercress aside. Heat oil in a medium skillet over medium heat. Add watercress leaves and green onions. Cook, stirring, until very tender, 3 to 4 minutes. Set aside to let cool.
In a food processor, combine yolks, watercress and green onion mixture, mayonnaise, mustard, salt and cayenne. Pulse until very smooth, about 1 minute, stopping to scrape down sides of bowl once or twice. Spoon yolk mixture into egg halves, place on a platter and surround with a few of the reserved watercress sprigs.
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