Heat the olive oil in a large heavy-bottomed pot over medium-high heat. Add the ground lamb and cook, breaking it up with a spoon, until it starts to turn light brown (about 5 minutes). Add the onion, garlic, and grated carrot, and sauté until the onions are translucent (about 3–4 minutes).

Stir in the tomato paste, paprika, dried basil, salt, and pepper. Cook for 1–2 minutes until fragrant.

Add the cubed potatoes and enough water to just cover them (approximately 1–2 cups / 240–480ml). Bring to a simmer, cover, and cook for 15–20 minutes until the potatoes are nearly fork-tender but still hold their shape. Stir in the chopped parsley.

In a medium saucepan, melt the butter over medium-low heat. Whisk in the flour and cook for 1–2 minutes, stirring constantly — do not let the roux brown. Remove from the heat and slowly pour in the milk in a thin stream, whisking continuously to prevent lumps. Return to low heat and simmer, whisking frequently, for 3–4 minutes until the sauce thickens enough to coat the back of a spoon. Remove from the heat. Spoon a tablespoon of the hot sauce into the egg yolks and whisk to temper, then stir the tempered yolks into the saucepan. Season with salt and nutmeg.

Brush a 9x13 inch (33x23cm) baking dish with olive oil. Transfer the lamb and potato mixture into the dish, spreading it evenly. Pour the béchamel sauce over the top, tilting the dish gently to ensure it fills the corners and covers the surface completely. Bake at 350°F / 175°C (160°C fan-forced) for 40–45 minutes until the top is golden brown and the edges are bubbling. Let it rest for 10–15 minutes before slicing to allow the layers to set.