Trim the ends off the zucchini and cut into noodles according to the instructions on your spiralizer.

Line two rimmed sheet pans with thick layers of paper towels. Spread half of the zucchini noodles onto each pan and sprinkle with the salt. Cover with another thick layer of paper towels and press gently. Leave noodles to dry while you prepare the rest of the dish.

Warm olive oil over medium heat in a very large pan or wide, shallow pot. (A 5-quart Dutch oven, a 12-inch frying pan with high sides, or even a wok would all work well.)

Add shallot, garlic, and capers, and cook, stirring frequently, for two minutes.

Stir in tomatoes and cook, stirring occasionally, until most of them have burst. 

While tomatoes cook, cut the kernels from the corn cobs. Add corn kernels and chickpeas to pan and cook for about two minutes, until warmed through.

Pat zucchini noodles dry with the paper towels and stir noodles into the vegetable mixture in the pan. (Tongs work really well if you have them.) Cook until noodles are crisp-tender, just a couple of minutes. The tomatoes and zucchini will create a bit of delicious sauciness in the bottom of the pan. Stir this to coat the noodles.

Remove pan from heat.

Cut slices of prosciutto into long, thin strips, about four per slice. Sprinkle it over the vegetables, along with the basil, pepper, and grated cheese. The residual heat from the noodles will warm the prosciutto just enough to be beautifully tender and flavorful, without cooking it. 

Divide among wide, shallow serving bowls to serve.