Peel the potatoes, cut them into smaller pieces, and boil in salted water for about 20 minutes, or until tender.
Drain the cooked potatoes, mash them in a bowl, and let them cool completely.
Add the flour, egg yolk, and a pinch of salt to the cooled potatoes. Knead until you have a soft, smooth dough.
Divide the dough into four parts. Roll each part into a rope about 2 cm thick, then cut into small pieces. Gently press each gnocchi with a fork to create the characteristic ridges.
Bring a large pot of water to a boil. Add the gnocchi in smaller batches. As soon as they float to the surface (about 2 minutes), drain them and set aside.
Preheat your grill to medium heat and brush the grates with olive oil.
Cut the zucchini, eggplant, and bell pepper into thick slices. Brush with olive oil and season with salt and pepper.
Place the vegetables on the grill and cook for about 3-4 minutes per side, until lightly charred and caramelized.
Brush the cooked gnocchi with a little olive oil and place them on the grill grates. Grill for 2-3 minutes, until light grill marks appear.
Before serving, combine the grilled gnocchi and vegetables. Sprinkle with grated Parmesan cheese and fresh basil.
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