Combine ketchup ingredients in a small pot and bring to a bubble. Reduce heat to low and simmer to thicken to the consistency of ketchup, 15-20 minutes.
Meanwhile, heat a tablespoon EVOO in a skillet over medium to medium-high heat. Add carrot, onions and mushrooms, and saut to tender. Add thyme and garlic, season with salt and pepper, cook a minute or 2 more then turn off heat and cool.
To a food processor bowl, add a few pieces of dried mushrooms. Process to finely chop then add lentils, quinoa or couscous, chickpeas, nuts and raisins. Season with salt and pepper and add cooled, cooked vegetables. Pulse until well combined. Add egg and pulse a few more times. Form 6 patties.
Heat remaining 2 tablespoons EVOO in a large skillet over medium-high heat. Add patties and cook 5-6 minutes. Flip burgers, cook 2-3 minutes more, add cheese and tent pan with tin foil to melt cheese, 1-2 minutes. Serve on rolls with greens and Italian ketchup.
For a vegan burger, omit egg -- be gentle when flipping burgers so they don't fall apart -- and omit cheese.