Heat olive oil over medium heat in a large pot. Cook and stir minced onion, garlic, and ginger in the hot oil until lightly browned, 5 to 8 minutes. Add potatoes, carrots, and chopped onions; stir until softened, about 5 minutes.

Pour water into the pot and bring to a boil. Skim off any residue that comes to the surface. Stir in laurel leaves. Reduce heat to medium and boil until all vegetables are easily pierced with a fork, about 10 minutes.

Break curry paste into blocks and stir 5 1/4 ounces into the pot. Add tomato, apple, and honey; stir to combine. Cover, reduce heat to low, and let simmer for 30 minutes. Stir in remaining pieces of curry paste if end mixture is too watery. Serve with cooked rice.