Cook rice with the amount of water called for in the package directions, adding scallions, garlic and thyme to the water. Remove from heat and stir in corn, breadcrumbs, salt and pepper. Let stand for about 5 minutes to cool slightly.
Whisk eggs in a large bowl until frothy. Add the rice mixture and mash with a potato masher until the mixture holds together (it will be fairly soft), about 1 minute. Shape the mixture into 8 patties.
Heat 2 teaspoons oil in a medium nonstick skillet over medium-high heat. Carefully transfer 4 patties to the pan with a spatula and cook until browned and crispy, about 3 minutes per side. Transfer to a plate; cover to keep warm. Wipe out the pan with a paper towel, add the remaining 2 teaspoons oil and repeat with the remaining 4 patties.
Meanwhile, combine beans and salsa in a small saucepan and cook over medium-high heat, stirring occasionally, until hot, about 5 minutes. To serve, divide the beans among 4 plates and top with 2 patties each.