Heat the vegetable oil in a large pot or deep fryer to 350°F.
Carefully add the cut potatoes in batches and fry until golden brown and crispy, about 5-7 minutes. Remove and drain on paper towels. Season with salt and pepper.
In a heatproof bowl, whisk together the egg yolks and lemon juice for the hollandaise sauce.
Place the bowl over a pot of simmering water (double boiler) and whisk continuously until the mixture thickens, about 3-5 minutes.
Slowly drizzle in the melted butter while whisking until fully combined. Season with salt and a pinch of cayenne pepper, if desired. Keep warm.
In a separate pot, bring water to a gentle simmer and add the white vinegar.
Crack each egg into a small bowl and gently slide them into the simmering water. Poach the eggs for about 3-4 minutes, or until the whites are set but the yolks remain runny. Remove with a slotted spoon and drain on paper towels.
To assemble, place a generous portion of fries on a plate, top with cheese curds, and then add a poached egg on top. Drizzle with hollandaise sauce and serve immediately.