Add the water, olive oil, bay leaves, and salt to the pot. While cold, whisk in the polenta slowly, pouring it in a thin stream. Raise the heat to medium and bring it to a simmer. Simmer for 30 minutes, stirring often to keep it from sticking.
Remove it from the heat and stir in the butter and parmigiano until well combined. While still hot, pour the polenta into a greased baking pan. Cover and refrigerate for at least 4 hours or overnight.
Preheat the grill to medium. Once chilled, the polenta should feel somewhat rubbery. Use a knife to cut your desired shapes into the sheet of polenta and set them on a plate. Lightly brush the polenta on both sides with olive oil.
Grill the polenta on each side for 5-6 minutes or until it easily comes off the grill grates. If it's sticking, it's not done cooking yet. It will unstick when it's done cooking.
Remove from the grill. Serve over marinara with fresh pesto and extra parmigiano cheese.
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Notes
Chill the creamy polenta for at least 4 hours. It will fall right through the grill grates if you don't chill for long enough. It needs time to set.
Do not try to scrape the polenta off the grill if it is sticking. If it's still sticking, it's not ready to be flipped.
The best way to reheat is to place it back on the grill.