Heat the olive oil in a 28 cm / 11-inch deep sauté pan over medium heat. Add the onion and cook for 5 to 6 minutes, stirring now and then, until soft and just starting to turn golden. Add the garlic and stir for 30 seconds until fragrant.
Stir in the tomato paste and cook for 1 minute until it darkens slightly. Add the carrots and potatoes and stir for 2 minutes so they're coated in the oil.
Pour in the crushed tomatoes and water, then add the sugar, salt and pepper. Bring to a boil, which takes about 3 minutes. Cover, turn the heat to medium-low and simmer for 10 minutes.
Add the frozen peas straight from the freezer and stir. Cover and simmer for 12 to 15 minutes, until a fork slides easily into a potato and the peas are soft. Adding the peas now, not at the start, keeps them from turning gray and mushy. If the sauce looks thin, uncover the pan for the last 3 to 5 minutes.
Turn off the heat. Stir in the dill and lemon juice, then taste and add salt a pinch at a time if needed. Serve warm or at room temperature with a drizzle of olive oil.