Thoroughly rinse the dried borlotti beans under cold water. Place them in a large bowl and cover them with water. Allow the beans to soak overnight or for at least 8 hours. Drain and rinse the beans again in a colander or wire mesh strainer.
Place the beans into a large pot with water to cover. Bring the pot to a boil over high heat. Once boiling, cover the pot and reduce the heat to medium and cook for about 40 minutes until the beans are tender. Drain the cooked beans in a colander or wire mesh strainer.
In a thick-bottomed cooking pot or a dutch oven, heat the extra virgin olive oil over medium heat. After 60 seconds when the oil is hot, add the finely chopped onion and sauté until it turns translucent and slightly golden, about 5-6 minutes. Stir in the minced garlic and cook for an additional 1-2 minutes until fragrant.
Stir in the tatli biber salçası and sauté for another minute to combine and allow the pepper paste to caramelize a little. If you want a spicier finished dish you can substitute acı biber salçası in place of the mild pepper paste.
After a couple of minutes when the paper paste becomes fragrant, add the thinly sliced carrots and tomatoes to the pot and cook for about 3-4 minutes, allowing the tomatoes to break down a bit.
Add the cooked borlotti beans to the pot. Sprinkle in the sugar and salt. Pour in 2 cups of water or unsalted vegetable stock to ensure the beans are well-covered. Bring the mixture to a gentle boil, then reduce the heat to low, cover the pot with a lid, and let it simmer and reduce for 20 minutes.
Once the beans are tender and the other ingredients have transformed into a thin sauce, remove the pot from the heat. Taste and adjust the seasoning if necessary. Serve the Barbunya Pilaki hot, garnished with thinly sliced half lemon wedges and fresh chopped parsley leaves.