Butter a 9×13-inch baking dish, and set it aside.
In a large Dutch oven or stockpot, bring the water and milk to a boil. Add the polenta in a gentle stream, whisking as you pour. Reduce the heat to medium-low and continue to whisk the polenta until it has thickened, about 10 minutes. Add the butter and salt, and continue to whisk until most of the liquid has evaporated, about 3 minutes. Pour the polenta into the prepared baking dish, and smooth it with a spatula to make an even layer. Cover with plastic wrap and chill in the refrigerator for 2 hours or up to 2 days.
Preheat the oven to 450°F Cut the polenta into 12 rectangular “steaks,” and pat each one dry with a paper towel.
Brush both sides of each polenta steak with olive oil. Place the steaks on a baking sheet and bake in the oven for 30 to 40 minutes without turning, until the tops have browned.
Meanwhile, prepare the pesto: Bring a pot of heavily salted water to a boil. Blanch the asparagus in two batches until bright green and just tender, 1 to 2 minutes. Use tongs to transfer the asparagus to an ice bath, or put them in a colander and rinse under cold water. Drain.
Cut the asparagus into 1-inch pieces. Put 2 cups of the stalk pieces in a small food processor, reserving the tips. Add the mint, garlic, lemon juice, pine nuts, olive oil, and salt, and pulse. Add 1/3 cup water, and pulse until the mixture has the consistency of pesto. Mix in the Parmesan, and taste for seasoning.
Place the reserved asparagus tips in a mixing bowl, and toss with the tomatoes, burrata, olive oil, and salt.
To serve, place a warm polenta steak in the center of each plate, and top it with a generous dollop of pesto and a spoonful of the cherry tomato topping.