To prepare the farro, bring the water, farro, bay leaf, peppercorns, and salt to a boil in a 2-quart saucepan. Decrease the heat to maintain a simmer, cover, and cook until the grain is tender but still slightly chewy, 20 to 25 minutes. Remove the bay leaf, drain any remaining liquid, and set aside.
While the farro simmers, rinse the mussels under cold running water, brushing to remove sand and residue on the shells. Remove the beards (hairy clumps around the shell) with tweezers or a sharp knife. Discard chipped mussels. Tap any open mussels and discard if they don’t close. Set the cleaned mussels aside.
To make the stew, heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium heat until shimmering. Add the onion, carrots, celery, garlic, 1 teaspoon of the rosemary, the bay leaves, chile, and 1/4 teaspoon of the salt. Cook, stirring frequently, until the vegetables soften, 3 to 5 minutes. Increase the heat to medium-high, add 1/4 cup of the white wine, and cook until syrupy and the liquid is almost gone, about 2 minutes. Add the tomatoes, the water, the remaining 11/4 cups white wine, the pepper, and the remaining 1/4 teaspoon salt; bring to a boil. Cook, uncovered, at a lively simmer until the carrots are crisp-tender, about 5 minutes. Stir in the sugar.
Add the mussels and the farro together with the remaining 1 teaspoon rosemary to the pot and bring to a boil. Cover and steam over medium to medium-high heat, shaking the pot once or twice in between, until the mussels open, 2 to 3 minutes. Remove from the heat, and discard any unopened mussels.
To finish, add the lemon juice. Taste for salt and pepper and adjust. Drizzle the mussels with the olive oil and serve right away in deep plates, garnished with parsley and with lemon wedges on the side.