Cut onion into half moons. Cut carrot in half lengthwise, then across into slices. Cut green beans into 2-inch pieces. Dice tomato.
In a medium size skillet with a lid, over medium heat, heat 2 Tablespoons olive oil. When hot, add onion a cook for 3 minutes. You don’t want any browning, so turn heat down if onions start to brown.
Add carrot and cook for 4 minutes. Add green beans, tomatoes, and tomato paste, and cook 3 minutes, stirring. Add ⅓ cup olive oil, water, sugar, salt, and bring to a boil.
Turn down the heat so that the mixture simmers, and cover with lid. Simmer covered for 35 minutes. Check liquid level periodically, and add more water if ingredients are sticking.
By the end of cooking there should not be much water left, but a nice olive oil/tomato sauce. Taste and adjust seasoning for salt. Let cool, or put in refrigerator to cool off. Beans will last 3 days in the refrigerator. Serve cold or at room temperature, with crusty bread.
This dish tastes better the next day.
Sugar is traditional in zeytinyagli dishes, to provide a balance of flavors. If you don’t prefer to use sugar, use one more carrot instead, which provides some sweetness as well.