Lightly coat a sheet pan with olive oil and set aside. Bring the water, milk, and olive oil to a boil in a large pot. Add salt and pepper, and then sprinkle in the cornmeal, stirring continuously with a wooden spoon or heavy whisk to avoid lumps. Reduce the heat to medium and continue cooking, stirring frequently, for about 15 minutes, or until the polenta pulls away cleanly from the sides of the pot. Season, to taste, again with salt and pepper. Fold in the butter and Parmesan.
Turn the polenta onto the prepared sheet pan and smooth it out. Cover it with plastic wrap, and put it in the refrigerator until it is quite cold, preferably overnight. Using a 4-inch round pastry cutter, cut the cold polenta into 4-inch round cakes. Place the cakes on parchment paper, leaving them covered with plastic wrap until ready to serve.
Close to serving time, heat a large skillet over medium-high heat. Add enough olive oil to coat the bottom of the skillet, and let the oil heat. Dust the polenta cakes with flour, add them to the pan in batches, and saute until crispy and golden brown. Top each cake with 1 tablespoon Taleggio, and place under the broiler until cheese is melted and a little browned on top. Serve the polenta cakes with basil oil, smoked tomato sauce, and grilled vegetables, layering like free-form lasagna as desired.
Blanch the basil for 3 to 5 seconds in a large pot of boiling salted water. Using a spider or a slotted spoon, immediately plunge the basil into a bowl of ice water to retain its bright green color. Gently squeeze the basil to remove as much excess water as possible.
Put the basil in a food processor or blender. Add both oils, and puree until smooth. Put the pesto in an airtight container, and refrigerate it overnight. Then pour off the excess oil that has settled on top. Pour pesto into squirt bottles for easier service.
Smoke the onions and fresh tomatoes in a smoker under mesquite chips. Remove from smoker, and set the tomatoes aside. Heat the olive oil in a large saucepan over medium-high heat. Add the smoked onions and garlic to the pan, and saute until tender and very fragrant. Add the oregano and bay leaves and cook for 1 more minute, or until fragrant. Add the smoked tomatoes and the canned tomatoes, and bring mixture to a boil. Add the white wine and basil, reduce the heat, and simmer for about 15 to 20 minutes. Using an immersion blender, puree the sauce until smooth. Then add the sugar, and simmer for another 10 to 12 minutes. Strain sauce through a medium-sized mesh sieve. Season to taste with salt and pepper.
Preheat a grill, or heat a grill pan over medium-high heat. In a large bowl, toss all of the sliced vegetables with the olive oil. Add salt and pepper, to taste. Add the garlic and fresh herbs, if using, and toss all together. Grill the vegetables until they are tender yet still slightly crisp.
* When cooking the cornmeal, it is difficult to assess the exact quantity of liquid which will be required, since flours differ in coarseness, plus the humidity and moisture level of the kitchen is a factor in determining the proper ratio of cornmeal to liquid.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.