Wash the green beans and trim both ends. If using flat Romano beans, leave them whole or snap in half if very long. For regular green beans, snap or cut into 5–6 cm (2-inch) pieces.

In a wide, heavy-bottomed pot or Dutch oven, warm the olive oil over medium heat. Add the chopped onion and cook, stirring occasionally, for 6–7 minutes until soft and translucent — not browned. Add the garlic and cook for another minute until fragrant.

Stir in the tomato paste and cook for 1–2 minutes until it darkens slightly and becomes fragrant. Add the grated fresh tomatoes or crushed canned tomatoes. Stir well, season with salt, pepper, and sugar if the tomatoes taste sharp. Let the sauce simmer for 5 minutes to concentrate.

Add the green beans and potato quarters to the pot. Toss to coat everything in the sauce. Add enough warm water to almost cover the vegetables — about 250 ml (1 cup). The beans should be mostly submerged but not drowning.

Bring to a gentle boil, then reduce heat to low. Cover the pot with a lid slightly ajar and let the fasolakia simmer for 50–60 minutes, until the beans are very tender and the sauce has thickened and reduced. The oil will rise to the surface and turn a deep orange-red — this is exactly what you want.

Remove from heat and let the fasolakia rest for at least 15–20 minutes. Serve warm in shallow bowls. Drizzle with fresh olive oil, scatter with parsley, and place a generous slab of feta alongside. Tear off some crusty bread and let it soak up the sauce.