Preheat the oven to 325 degrees F. Position the oven rack in the upper 1/3 of the oven.
On a rimmed baking sheet, combine the olive oil, mushrooms, chickpeas, thyme, garlic, and a pinch of salt., pepper and crushed red pepper flakes.
Transfer to the oven and bake for 25-30 minutes. Increase the heat to 475 degrees F. Toss the mushrooms and chickpeas, then return to the oven and continue to cook for another 5-10 minutes until crisp on the edges. Watch closely.
Meanwhile, make the polenta. In a medium saucepan, bring 4 cups water to a boil over medium heat. Slowly whisk in the polenta, stirring until the polenta is soft and thick, about 5 minutes. Stir in the butter with olive oil, kale or spinach and 2 tablespoons pesto, stirring until the spinach or kale has wilted, about 3 minutes. If the polenta thickens too much, thin with water.
Divide the polenta between bowls and break the burrata over the polenta. Spoon the mushrooms and chickpeas over the polenta. Drizzle with additional pesto and garnish with fresh thyme.