Coarsely grate potatoes on a box grater. Spread in an even layer on a large kitchen towel. Roll the towel into a log. Over the sink, wring the towel to release as much moisture as possible. Transfer the potatoes to a large bowl. Season with salt.
Heat about 2 tbsp (30 mL) oil in a large skillet set over medium-high heat. Dollop the potato mixture into the skillet into four ¼ cup (60 mL) portions. Gently press down to flatten. Cook, without disturbing, for 4 to 5 min. per side or until golden. Transfer to a paper towel-lined plate. Repeat with remaining potato mixture, adding extra oil if necessary.
Meanwhile, heat 1 tbsp (15 mL) in another large skillet set over medium-high heat. Cook, burgers in batches, for 4 to 5 min. per side or until cooked through, and internal temperature reaches 160°F(71°C), adding more oil as needed. Transfer burgers to plate. Keep warm.
Wipe out one of the skillets. Melt butter in it over medium heat. Add shallots and thyme and cook 2 to 3 min., or until softened. Sprinkle in flour and cook, stirring, 2 min. or until foaming. Slowly whisk in broth. Bring to a boil then reduce to a simmer 5 to 6 min. or until thickened. Stir in pepper to finish gravy.
Place burger patties on buns. Top with grated potato rounds, gravy, cheese curds and chives.
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