Heat 1 Tablespoon butter over medium-high heat in a skillet; turn down to medium, and cook onion, stirring occasionally, about 15 minutes or until golden brown. Set aside to cool.
Add wine and continue cooking til absorbed. Add garlic and thyme, stirring, about 2-3 minutes. Remove from heat, and let cool.
Combine beans, Worcestershire Sauce, breadcrumbs, egg, rice, oatmeal, carrots, salt, and pepper in processor and processor til still slightly chunky.
Add the onion mixture and mix with a spoon until well-combined. Form mixture into 6 patties (about 1/2 cup each). These are best if they are not super fat patties, thinner is better. Put cornmeal on a shallow plate, and coat burgers evenly.
Heat a thin layer of oil in a nonstick skillet over medium heat, and cook burgers about 5 minutes or until golden brown on underside.
Flip burgers, and top with cheese. Cook about 5 minutes more, partially covered, or until golden brown and cheese is melted. Serve burgers on arugula and top them with garlic aioli and caramelized onions.