Preheat the oven to broil. In a deep-fryer or heavy-bottomed pot, heat enough canola oil to come halfway up the sides of the pot, to 375 degrees F.
Slice the potatoes into 1/4-inch thick french fries. Fry in batches until golden brown and crispy; about 5 minutes. Drain on paper towel lined sheet tray and put them in a large baking dish. Season with salt and pepper, to taste.
Melt 2 tablespoons of the butter in a medium-sized saucepan over medium-high heat. Stir in the flour and the garlic. Add the wine and beef stock and bring to a simmer.
In a small bowl, combine the water and cornstarch. Slowly whisk the mixture into the stock and cook until thickened, roughly 2 to 3 minutes. Add salt and pepper, to taste. Remove the pan from heat, and stir in the remaining 2 tablespoons butter.
Ladle the gravy over the fries and sprinkle with the cheese. Put the fries in the oven and bake until the cheese is melted and bubbling. Remove the fries from oven, sprinkle with chopped chives and serve.