For a richer flavor, chef Uri Navon of Machneyuda restaurant cooks his polenta in heavy cream instead of in water.
Make the mushroom ragout: Heat butter in a small heavy-bottomed skillet. Add mushrooms and cook, stirring occasionally, until thoroughly cooked, 25 to 30 minutes. Season with salt and pepper.
Meanwhile, make the polenta: Bring milk and heavy cream to a boil in a large saucepan over medium high-heat. Reduce heat to low and whisk in polenta; cook, whisking constantly, until thickened, 5 to 7 minutes. If mixture seems too thick, thin out with a little more milk or water. Whisk in butter and Parmesan cheese. Season with salt; cover and set aside.
Make the asparagus: Bring a large pot of water to a boil; generously salt water and return to a boil. Meanwhile, prepare and ice-water bath. Add asparagus to boiling water and cook until tender-crisp, 1 to 2 minutes. Drain and transfer to ice-water bath. Drain and transfer to a medium bowl. Drizzle with olive oil and lemon juice; season with salt and pepper.
To serve, place 2 to 3 tablespoons polenta in each of four 1-cup jars with lids. Add 1 tablespoon mushroom ragout to each jar. Top with 3 to 4 asparagus pieces and 2 to 3 Parmesan shavings each. Drizzle each jar with truffle oil; cover jars and serve.