For the quiche: Preheat the oven to 350 degrees F. Put a 10-inch square baking dish on a baking sheet, brush the bottom and sides with the butter and sprinkle with the toasted cornmeal.
Heat 2 tablespoons of the canola oil in a large skillet over high heat until it begins to shimmer. Add the shallots and cook, stirring, about 1 minute, then add the garlic and cook, stirring, another minute. Add the mushrooms and cook, stirring occasionally and adding up to 1 tablespoon more of oil if necessary, until lightly golden brown and their liquid has evaporated, about 10 minutes. Add the parsley and thyme. Season with salt and pepper. Add the wine and cook until almost all of the liquid is gone, about 1 minute. Remove to a plate and cool.
Bring the milk and 3 cups water to a boil in a large saucepan over high heat. Add 1 tablespoon salt and 1/4 teaspoon black pepper. Slowly whisk in the grits and bring to a boil, stirring. Cook the grits, stirring occasionally, until thickened and slightly soft, about 15 minutes. Remove from the heat and stir in the Asiago and half the Cheddar and some salt and pepper.
Whisk the eggs in a large bowl. Whisk in a small amount of the grits mixture to warm the eggs, Whisk in the rest of the grits mixture and then fold in the mushrooms. Transfer the mixture to the prepared baking dish and top with the remaining Cheddar. Bake the quiche until it is just set and the cheese is bubbling, about 50 minutes. Cool 10 minutes before serving.
For the pesto: Combine the kale, Parmesan, walnuts, garlic, salt and pepper and a splash of the olive oil in a food processor and pulse several times to coarsely chop. With the motor running, add the remaining oil until the pesto is thick and creamy. Stir in the lemon zest and season with some salt and pepper. If too thick, add a few tablespoons of cold water.
To serve: Spoon the quiche into shallow bowls. Spoon some of the kale pesto on top and drizzle with olive oil. Serve warm with the hot sauce.