Make the gnocchi: In a medium saucepan, bring the butter, water, milk, and 1½ teaspoons of salt to a simmer over medium-high. Add the flour and cook over low, stirring constantly, until a dough forms and a film develops on the bottom of the pot, 3 to 5 minutes.
Transfer the dough to a mixing bowl and stir with a spatula to cool it down slightly. When no longer steaming, add the eggs one at a time, stirring vigorously to fully incorporate each egg, until the dough is smooth and glossy. Spoon the dough into a piping bag fitted with a 3/4-inch round piping tip.
Meanwhile, tie a length of thin kitchen twine tightly across a large two-handled Dutch oven. Fill the pot halfway with water and bring to a simmer over medium-high. Add a generous pinch of salt, then reduce the heat to medium-low to maintain a gentle simmer.
Using steady pressure and carefully working over the pot of simmering water, pipe one-third of the gnocchi dough into 1-inch lengths, using the twine to cut the gnocchi into the saucepan. Simmer until the gnocchi rise to the surface, then continue simmering until the gnocchi are plump and cooked through, about 5 minutes total. Using a slotted spoon, transfer the gnocchi to a lightly oiled rimmed baking sheet. Repeat the process with the remaining 2 batches of dough.
Preheat the oven to 375°F. In a 3- to 4-quart ovenproof braiser, heat 1 tablespoon of olive oil over medium. Add the spinach and kale mixture and a generous pinch of salt. Cover and cook, stirring occasionally, until the greens are wilted, about 5 minutes. Transfer to a colander to drain and cool, then squeeze out the excess water and coarsely chop.
Wipe out the braiser and melt the butter in the remaining 2 tablespoons of olive oil in it. Add the shallot and garlic and cook over medium, stirring occasionally, until softened, about 5 minutes. Sprinkle the flour and top and cook, stirring, until bubbling and golden, 3 to 5 minutes. Gradually whisk in the milk until smooth. Simmer over low, stirring, until thickened and no floury taste remains, about 5 minutes. Stir in half the Comté until smooth. Stir in the nutmeg and season with salt and pepper.
Gently fold the greens into the sauce, followed by the gnocchi. Sprinkle the remaining Comté and the Parmesan on top. Bake in the preheated oven until bubbling, about 20 minutes. Switch on the broiler and broil until lightly browned, about 1 minute. Let cool for 10 minutes before serving.
