Preheat the oven to 230°C (450°F) with racks in the upper and lower thirds. Line two baking sheets with parchment.
Cut the potatoes into thick fries about 1.5 cm (1/2 inch). Soak them in cold water for 10 minutes to remove surface starch, then drain and dry very well with a towel.
Toss the fries with the oil and salt and spread in a single layer on the two trays, not touching. Bake for 30 to 35 minutes, flipping the fries and swapping the trays at 15 minutes, until golden and crisp.
If the chicken is raw, poach it: cover the breasts with water in a pot, bring to a simmer over medium heat and cook gently for 12 to 15 minutes, until 74°C (165°F) inside. Shred with two forks.
For the gravy, bring the broth, Worcestershire sauce, ketchup and pepper to a simmer in a saucepan over medium heat. Mix the cornstarch with 3 tbsp of cold water.
Whisk the cornstarch mixture into the simmering broth and cook, whisking, until thick and glossy, about 2 minutes.
Pile the hot fries into bowls, top with the warm chicken and the cheese, and pour the hot gravy over so the cheese softens. Scatter with green onion and eat right away.
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Tips
Real Quebec cheese curds squeak; they are worth finding at most grocery stores.