Heat a large saute pan over medium heat. Add the bacon, onion, and garlic and saute until onion is translucent, about 10 minutes. Add the shiitakes, chanterelles, oyster mushrooms, and rosemary and saute for another 4 minutes. Remove from the heat and set aside.
Combine the grits, milk, 1/2 teaspoon of the salt, and 2 cups water in a medium saucepan and bring to a simmer over low heat. Simmer for 15 minutes, stirring occasionally. Remove from the heat, cover, and let sit for another 10 minutes.
While the grits are cooking, poach the eggs by bringing 4 cups water to a boil in a large heavy-bottomed saucepan. Add the vinegar and the remaining 1/2 teaspoon salt and reduce the heat to a very low simmer. Break 1 egg into a small cup or ramekin and gently slide the egg into the water. Repeat with the remaining eggs. Poach until the whites are firm and translucent, 2 to 3 minutes. Remove with a slotted spoon, strain, and transfer to a bowl. Set aside.
In the bowl of an electric mixer fitted with a whisk attachment, whip the cream until soft peaks form. Fold the whipped cream, mushroom mixture, olive oil, Parmesan, truffle oil, basil, and pepper into the grits. To serve, place each egg in a bowl. Spoon the grits over the top.