For the pomegranate molasses: In a small saucepan, bring the pomegranate juice, maple sugar and lemon juice to a slight boil over medium-high heat. Reduce the heat to low and simmer until syrupy, about 30 minutes. Remove from the heat and set aside.
For the burgers: In a large pan, heat 1 tablespoon oil over medium heat. Add the onion and cook until softened, about 5 minutes. Add the mushrooms, carrot and celery and cook, stirring occasionally, until the liquid from the mushrooms has evaporated, about 10 minutes. Transfer to a large bowl and let cool about 10 minutes. Stir in the rice, oats, walnuts, flour and 1/2 teaspoon salt. Transfer to a food processor and pulse until coarsely combined and the mixture just holds its shape when pressed, adding water, 1 tablespoon at a time, if necessary. Shape the mixture into 4 patties.
Stir together the ketchup with 2 tablespoons of the pomegranate molasses and set aside. Heat the remaining tablespoon olive oil in a large pan over medium-high heat and cook the patties until golden, turning once, about 8 minutes total. Arrange 4 English muffin halves on plates, if using, and top with some pomegranate ketchup, the patties, spinach leaves, carrots, pomegranate seeds, more ketchup and the remaining English muffin halves.