Mix rice with water and a dash of salt in a 4-quart saucepan; bring to a boil. Stir, cover, reduce heat, and let simmer until tender, 40 to 45 minutes.
At the same time, drain most of the liquid from the beans and combine with onion and cayenne in a small saucepan. Bring to a simmer and cook, covered, 15 to 20 minutes. Add cilantro and lime juice, stir, and remove from heat.
Combine garlic and salt for sauce in a small bowl; mash together to form a paste. Add garlic paste to sliced onion, olive oil, lemon juice, orange juice, and cumin in an electric blender and process until smooth. Transfer to a small nonstick skillet and simmer, stirring regularly and taking care not to burn, for 5 minutes.
Layer 1 cup of rice, 2 to 3 tablespoons mojo Cubano, 3/4 cup black beans, a generous sprinkling of Cheddar cheese, salsa, and a dollop of sour cream in a large bowl. Repeat for each serving.