Sauté the potatoes. Heat a generous drizzle of olive oil in a large skillet or Dutch oven over medium-high heat. Add the potatoes, season with salt, black pepper, and paprika, and sauté for several minutes until they begin to develop golden edges. They don't need to be fully cooked yet.
Add the green beans. Add the trimmed green beans to the potatoes and toss everything together so the vegetables are coated in the seasoned olive oil. Cook for a few minutes, allowing the beans to pick up some color while the potatoes continue to cook.
Build the tomato base. Stir in the chopped tomatoes and tomato paste, then add the Better Than Bouillon, nutritional yeast, sugar, and another pinch of salt. Stir well so the tomato paste coats the vegetables and starts to cook into the olive oil.
Add the liquid and simmer. Pour in the water or vegetable stock, using enough to create a generous tomato sauce without completely submerging the vegetables. Bring everything to a gentle simmer, cover, and cook until the potatoes and green beans are tender, stirring occasionally and adding another splash of liquid if needed.
Reduce the sauce. Once the vegetables are tender, uncover the pan and let everything simmer for a few more minutes if the sauce needs to thicken. Taste and adjust the salt, paprika, pepper, or sugar. You want the finished sauce to be rich and tomatoey, with enough olive oil to give it that signature lathera-style finish.
Finish and serve. Drizzle generously with good olive oil before serving. Enjoy the Fasolakia Lathera warm with crusty bread, rice, or a simple salad, or let it become part of a larger spread of naturally vegan Greek-inspired dishes.