Step 1

In a medium bowl, stir together the milk, polenta and salt. Cover and refrigerate overnight.

Step 2

The next morning, set up a double boiler with a heatproof bowl over a simmering pot of water. Transfer the soaked polenta and milk to the heatproof bowl. Cook, whisking constantly, until the polenta has thickened and is tender, 25 to 30 minutes.

Step 3

Turn off the heat and whisk in 2 tablespoons of the butter. Cover with plastic wrap, placing the plastic directly over the polenta to prevent a skin from forming and keep warm.

Step 4

Heat a cast iron skillet over medium high heat. Add remaining butter and the olive oil, followed by the mushrooms, thyme and chile. Season with salt and cook until golden brown and tender, 8 to 10 minutes. Adjust for seasoning.

Step 5

Divide the polenta between two bowls and top each with the sautéed mushrooms. Place a poached or fried egg over each, then serve.