Make the gnocchi dough. Place 500g ricotta, 1 large egg, ¼ cup grated Parmesan, 1 tsp Kosher salt, and ¼ tsp black pepper in a large bowl, then stir until well mixed. Add 140g all-purpose flour, then fold with a rubber spatula until the dough just comes together – it will be sticky. Lightly flour a work surface, then dump the gnocchi mixture on top. Knead gently with your hands to form a mostly smooth ball, then set aside for 5-10 minutes, allowing the flour a chance to be absorbed. If the dough is still too sticky to work with after 10 minutes, add flour 1 tablespoon at a time until you can easily handle the dough.
Form the gnocchi. Line a baking sheet with parchment paper, then dust liberally with flour. Using a knife or a bench scraper, cut the gnocchi dough into 4 equal pieces. It’s more manageable this way. Roll each piece into a log about ½-inch thick, adding more flour to the work surface if it begins to stick. Use a knife or bench scraper to cut each rope into ¾-inch pieces. Immediately place on the floured baking sheet.
Prepare the water. Bring a large pot of water to a rapid boil, then season generously with 1-2 tablespoons Kosher salt. Meanwhile, make the sauce.
Prepare the browned butter sauce. Melt 4 Tbsp unsalted butter over medium-high heat, then add shucked corn and tomatoes. Season with ½ tsp Kosher salt, then cook, stirring occasionally, until the corn is lightly browned and the tomatoes begin to burst, about 6-8 minutes. Add 2 cloves chopped garlic, then cook 1 minute more. You can use the spatula to help break up the tomatoes, if wanted. Reduce the heat to low.
Cook the gnocchi. Carefully add gnocchi to the boiling water, then cook until they float to the top, about 2 minutes. Using a slotted spoon or spider, carefully transfer the gnocchi to the sauce. Reserve the cooking water.
Finish in the sauce, then serve. Raise the heat to medium-high, then cook the gnocchi in the sauce for about 2-3 minutes. Add ⅓ cup of cooking water and ¼ cup of grated Parmesan, then gently toss the gnocchi in the sauce. Serve in individual bowls with the sauce spooned on top along with additional grated Parmesan, fresh basil leaves, a generous pinch of flaky sea salt, and lots of freshly ground black pepper.