Simmer the polenta. Combine the polenta or cornmeal, water, and Kosher salt in a Dutch oven and set over medium-high heat. Bring to a simmer, stir, and reduce the heat to low. Cook, stirring often to prevent the polenta from sticking to the bottom of the pot, until the grains are fully cooked, about 40 minutes.
Sauté the mushrooms. While the polenta cooks, heat olive oil in a large skillet over medium-high heat. Add the mushrooms, shallot, garlic, thyme, rosemary, and pepper. Cook, stirring frequently, until mushrooms release some of their liquid and begin to shrink, 10 minutes.
Add the wine. Pour in wine and use a wooden spoon to scrape up the browned bits on the bottom of the pan. Cook, stirring, until the wine has nearly evaporated, about 2 minutes.
Add the greens. Add salt and baby spinach. Stir gently until wilted. Season with additional salt and pepper if needed
Finish the polenta. When the polenta is fully cooked, turn off the heat. Stir in butter and Parmigiano-Reggiano cheese until completely incorporated and creamy. Taste and add additional salt and pepper if desired. If you find it is too thick towards the end of cooking, stir in a little more water.
Serve. Spoon the hot polenta into shallow bowls and serve, topped with the mushroom mixture. Garnish with shaved Parmigiano-Reggiano and a drizzle of olive oil.
Notes
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I like to use medium grind or a combination of fine- and medium-grind cornmeal for the creamiest results.
Be careful! Polenta will bubble thickly, sending up splatters from time to time while cooking, especially when stirred. I recommend stirring it with a long-handled wooden spoon to keep your hands far away!
Adjust the texture: After 45 minutes, the polenta should be thick and creamy but still pourable. If you find it is too thick towards the end of cooking, stir in a little more water.