Melt 2 tablespoons butter in a medium skillet over medium heat. Add the onion, season with salt and cook, stirring occasionally, until golden brown and very soft, 25 to 30 minutes. Let cool.
Meanwhile, place the potatoes in a saucepan, cover with water by 1 inch and season with salt. Bring to a boil, then reduce the heat and simmer until tender, 10 to 12 minutes. Drain, then transfer to a large bowl, add the milk and mash until smooth. Let cool at least 30 minutes.
Preheat the oven to 250 degrees F and line a baking sheet with parchment paper. Add the cooked onion, egg yolk, flour, cheese, parsley, 1 teaspoon salt and a few grinds of pepper to the potatoes; stir until combined. Scoop 1/4 cupfuls of the potato mixture onto a lightly floured surface and form into twelve 3-inch pancakes.
Melt the remaining 1 tablespoon butter in a large nonstick skillet over medium-high heat. Working in batches, cook the pancakes until golden brown on the bottom, about 4 minutes. Gently flip, adding more butter to the pan if necessary, and continue cooking until browned on the other side, 3 to 4 more minutes. Transfer to the prepared baking sheet and keep warm in the oven until ready to serve. Top with chives.