Line a rimmed nine x 13-inch baking sheet with parchment paper and set aside.

In a pot, bring broth and milk to a boil. Whisk in cornmeal until smooth. Reduce heat to medium and cook eight to 10 minutes, stirring frequently.

Remove from heat and stir in butter, thyme, garlic, and salt. Immediately transfer to the prepared baking sheet. Even out the surface of the polenta until smooth; it should be about two inches thick. Cover with plastic wrap and chill about two hours to firm up.

Grease a second baking sheet. Slice polenta into strips and place on the baking sheet.

Heat oil in a large sauté pan over medium-high heat. Fry polenta strips in batches, cooking for five minutes per side or until golden. Alternatively, bake the polenta: Preheat oven to 450 degrees Fahrenheit. Drizzle oil on top and bake 15 minutes. Flip the strips and bake 10 to 15 minutes more, or until edges are crispy and browned.

In a large saucepan over medium-high heat, cook butter, shallots, and garlic until shallots are translucent.

Add mushrooms and cook another three to five minutes.

Add flour and butter, whisking to create a roux. Once fully incorporated, add wine and milk. Cook five to 10 minutes and top with parsley. 

Sous vide: Set up a water bath and place a sous vide precision cooker in it. Set the precision cooker to 167 degrees Fahrenheit. Gently place eggs in the water and cook 13 to 14 minutes. Make a small crack in the shell and the poached egg will slide out.

In a pot: Fill a saucepan about two-thirds full with water and bring to a boil. Add a splash of white vinegar and reduce the heat. Taking one egg at a time, crack the egg into a small cup. Create a gentle whirlpool in the water to help the egg white wrap around the yolk. Slowly tip the egg into the center of the whirlpool. Cook about three minutes and remove with a slotted spoon.

To serve, place polenta on each plate and top with mushroom ragu and a poached egg.