Prepare rice according to package directions. Preheat grill to medium-high heat, grease grates well.
Burgers: In large skillet set over medium-high heat, melt butter. Add onion, mushrooms and garlic and cook, stirring occasionally, for 8 to 10 minutes or until fragrant and tender. Stir in rice and breadcrumbs; set aside.
In food processor, combine egg, lentils, tomato paste, cumin, vinegar, tamari, chili powder and paprika until a paste forms. Season with salt and pepper. Transfer to medium bowl and combine with mushroom-rice mixture.
Divide into four portions. Shape into 1/2-inch-thick patties.
Grill patties, turning once, for 6 to 8 minutes or until char-marked and heated through. Top each with cheese slice and grill for 30 to 60 seconds or until melted.
Sauce: In small bowl, stir together mayonnaise, barbecue sauce, Dijon and pickles.
For serving: Serve patties in buns with sauce, arugula, tomato and onion.
Alternatively, in large cast-iron pan over medium-high heat, add 2 tbsp vegetable oil. Add patties and cook, turning once, for 6 to 8 minutes or until heated through.