Preheat the oven to 350 degrees F.
Put the mushrooms on a rimmed baking sheet and bake for 10 minutes. Let cool for 5 minutes, then transfer to a food processor. Add the beans, rice, corn, celery, lemon juice, shallot, cilantro, and coriander and process, slowly adding the broth until the mixture is well blended and chunky but not creamy. It should hold its shape if you roll a small portion into a ball. You may need to process the mixture in batches depending on the size of your food processor.
Line a baking sheet with parchment paper. Remove the blade from the food processor. Form the mixture into 16 equal-sized patties (use about 1/2 cup of the mixture for each patty) and put them on the lined baking sheet. Bake for about 15 minutes, until crisp and golden brown.
While the patties are baking, put the salsa, kale, and celery root in a large saucepan and bring to a simmer over medium-high heat. Decrease the heat to low and cook for 2 to 3 minutes. Divide the salsa mixture among four plates and top with the patties. Serve immediately.