Step 1

Whisk 2 1/2 cups all-purpose flour, 1 teaspoon baking powder, and 1/2 teaspoon kosher salt together in a large bowl. Create a deep well in the center. Whisk 1 large egg, 1/2 cup melted unsalted butter, 1/2 cup warm water, and 1 teaspoon distilled white vinegar together in a medium bowl. Pour into the well.

Step 2

Using a fork and starting in the center, gradually stir and pull in the flour mixture until a shaggy dough forms. Knead the dough a few times until it forms a smooth, soft ball. Wrap the dough tightly in plastic wrap and let rest for 1 hour to firm up. Meanwhile, make the filling.

Step 3

Place 1 1/2 pounds peeled and cut Yukon gold potatoes and 2 tablespoons of the kosher salt in a medium saucepan and add enough cold water to cover by about 1 inch. Cover and bring to a boil over medium-high heat. Uncover and reduce the heat to maintain a simmer. Cook until knife tender, 15 to 20 minutes. Drain in a colander.

Step 4

Melt 2 tablespoons unsalted butter in the same saucepan (no need to wash) over medium-high heat. Add 1 finely diced large yellow onion and 2 grated garlic cloves. Cook, stirring frequently, until the onions are translucent and lightly browned in spots, about 8 minutes.

Step 5

Add 1/4 cup water, scrape up any browned bits from the bottom of the saucepan, and cook until the water is almost completely evaporated, about 45 seconds. Remove the saucepan from the heat.

Step 6

Return the potatoes to the saucepan. Add 1 teaspoon spicy brown mustard and 1/4 teaspoon black pepper. Mix and mash with a fork or potato masher until the potatoes are broken down but still lumpy. Taste and season with more kosher salt and pepper as needed.

Step 7

Divide the dough in half (about 11 ounces each). Working with one piece at a time, roll out into an 8- by 16-inch rectangle (about 1/8-inch thick). Proceed with either baking or frying instructions.

Step 8

Heat the oven to 375ºF. Line a baking sheet with parchment paper. Spread half of the potato filling (about 1 1/2 cups) onto one of the dough rectangles in a log along one of the long sides. Starting with the end with the filling, roll up into a log. Repeat with the remaining dough and filling.

Step 9

Cut each log crosswise into 4 pieces. Pinch the dough along one cut side of each piece together to cover and seal in the filling on that side. Place sealed-side down on the baking sheet, spacing them evenly apart. Using the palm of your hand, gently press down until each knish is about 1 1/2 inches tall.

Step 10

Lightly brush the dough of each knish with 1 beaten large egg. Bake until golden brown, about 45 minutes.

Step 11

Cut each dough rectangle in half crosswise and lengthwise to get a total of 8 pieces. Divide the filling between the dough pieces (about 1/3 cup each). For each knish, fold the dough over the filling to form a square. Press the edges with your fingers or the tines of a fork to seal (make sure to press out any air). Trim the edges with a knife (save the scraps to test the temperature of the oil!).

Step 12

Heat 2 cups neutral oil in a medium cast iron skillet or Dutch oven over medium heat until shimmering (300 to 350ºF). Meanwhile, fit a wire rack over a baking sheet.

Step 13

Test the oil with a small piece of scrap dough: If it immediately starts bubbling, proceed with frying (if it doesn’t, the oil is not hot enough). Fry in batches of 2 to 3: Add to the hot oil and fry, flipping halfway through, until golden brown and heated through, 4 to 6 minutes total. Transfer to the rack. Let cool slightly before serving.