Mix the yeast and water in the bowl of a stand mixer until the yeast dissolves.
Add the flour and salt to the bowl, then use the dough hook to beat until a smooth, elastic dough forms (3 minutes or so). Place the dough in an oiled bowl and turn to coat. Cover the bowl with plastic and refrigerate at least 24 hours.
Divide the dough into 2 pieces and roll into balls. You'll only need 1 ball for this recipe - return the rest of the dough to the refrigerator or make a second pizza.
Place your baking stone or steel in the oven on middle rack and preheat oven to 500 degrees (250 C) for an hour.
Put a saute pan over medium heat and coat the bottom with olive oil. Add a pinch of salt and the onion to the pan and cook until softened. Toss in the garlic and mushrooms and cook until the water from the mushrooms escapes. Stir the mushrooms around and cook a minute or two more. Remove from the heat and stir in the kale. Set aside until ready to bake the pizza.
Place the ball of dough on a lightly floured counter and pat out to a circle with your hands. Flip the dough over to coat both sides with some flour. Pick up the dough and gently stretch, holding the edges of the dough while rotating it around – gravity will do the work. Fit your knuckles under both sides of the dough and stretch a little more to make a 12-inch round, keeping the edges just a bit thicker.
Set the dough on a floured baking peel or an upside-down baking sheet. Working quickly, drop the ricotta over the surface of the dough in spoonfuls. Top with kale and then the mushrooms.
Slide the dough (or gently use a pair of tongs) onto the baking stone or steel and bake until the edges of the dough begin to bubble and turn golden brown, about 8 minutes. Keep your eye on it, as oven temperatures can vary.
Drizzle the hot pizza with a little extra virgin olive oil and grate some Pecorino over before slicing and serving.
The pizza dough recipe makes enough dough for 2 12-inch pizzas - just double the amount of mushrooms, kale and ricotta as needed to make more than one pie.