Bring a medium sauce pot of generously salted water to a boil. Add the fusilli and cook until just tender, 8 to 10 minutes. Drain, reserving ⅓ cup pasta cooking water.
While the water heats and the fusilli cooks, cook the corn.

• Shuck the corn. Lay the cob flat and cut the kernels from the cob.

In a dry large frying pan over medium-high heat, add the corn and cook, stirring occasionally, until just tender and lightly charred, 3 to 5 minutes. Transfer to a plate. Wipe out the pan.
While the corn cooks, prepare the remaining pasta ingredients.

• Trim the ends from the summer squash; cut the squash in half lengthwise, then crosswise on the diagonal into ¼-inch-thick slices.
• Peel and finely chop enough shallot to measure 2 tablespoons.
• Finely chop, press, or grate enough garlic to measure 1 teaspoon.
• Cut the cherry tomatoes in half.
• Zest and juice half the lemon; save the other lemon half for another use.
• Strip the basil leaves from the stems; coarsely chop the leaves.
• Strip the parsley leaves from the stems; coarsely chop the leaves for garnish.

In the same pan used for the corn, warm 2 tablespoons oil over medium heat until hot but not smoking. Add the squash, season with salt, and cook, stirring occasionally, until lightly browned and starting to soften, 2 to 3 minutes. Stir in the shallot and garlic and cook until fragrant, 1 to 2 minutes.
Add the tomatoes, paprika, and as many chile flakes as you like. Cook, stirring occasionally, until the tomatoes start to break down, 2 to 3 minutes. Add the fusilli and reserved pasta cooking water and cook until hot and the liquid has thickened, 1 to 2 minutes. Remove from the heat. Add the corn, lemon zest and juice, and basil and toss to combine. Season to taste with salt and pepper.

Transfer the fusilli to individual bowls and crumble the ricotta salata on top. Garnish with the parsley and Parmesan and serve.