Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce the heat to medium-low and simmer until tender, about 20 minutes.

While the potatoes are cooking, heat oil in a large skillet over medium heat. Add onion; cook and stir until translucent, about 5 minutes, adding garlic for the last minute. Stir in diced tomatoes and tomato paste; cook until tomatoes soften and begin to lose their shape, 3 to 5 minutes.

Add ground beef to the skillet; cook and stir until browned and crumbly, about 10 minutes. Stir in Panquehue cheese, parsley, and cayenne pepper; season with salt and pepper. Turn off the heat and let sit until needed.

Preheat the oven to 350 degrees F (175 degrees C). Grease a 2-quart baking dish.

Drain potatoes in a colander, then allow to steam dry for 1 to 2 minutes. Transfer potatoes back to the pot, add butter and salt, and mash until smooth. Let cool until just warm, 5 to 10 minutes, then stir in beaten egg.

Spread 1/2 of the mashed potatoes over the bottom of the prepared baking dish. Layer ground beef mixture over top, then completely cover with remaining potatoes.

Bake in the preheated oven until potatoes are hot and lightly browned on top, about 40 minutes.