For the corn puree, combine corn, onion, oil, and a pinch of salt in a medium saucepan. Cook over low heat, stirring occasionally, until onion is softened and translucent, 6–7 minutes. Add 2 cups broth, increase heat to high, and bring to a boil. Reduce heat to medium-low and simmer until corn is softened and cooked through and liquid is reduced by half, 20–25 minutes. Let cool slightly. Transfer mixture to a blender and purée until smooth.
Bring 2 cups broth, farro, a pinch of salt, and 1 cup water to a simmer a large saucepan. Cook until farro is tender, 30–45 minutes, drain and return to pot.
Meanwhile, heat oil in a large skillet over medium heat. Add onion and cook until just beginning to soften, about 3 minutes and add farro.
Add corn purée to farro and cook over medium heat, stirring occasionally and adding more broth by 1/4-cupfuls if dry, until farrotto is very creamy, 5–6 minutes. Stir in cheese. Season with salt and pepper.
Divide into plates and add fresh tomatoes and basil on top.