Heat the vegetable oil in a large skillet over medium heat. Add the cubed tofu and cook until lightly browned on all sides, about 5-6 minutes. Remove the tofu and set it aside.
In the same skillet, add the diced onion and cook until softened, about 3 minutes. Stir in the minced garlic and grated ginger, cooking for 1 minute until fragrant.
Add the Thai red curry paste and stir to coat the onions. Pour in both cans of coconut milk and bring the mixture to a gentle simmer.
Add the broccoli florets, sliced red bell pepper, and carrot rounds. Simmer for 8-10 minutes until the vegetables are tender but still hold their shape.
Return the browned tofu to the skillet. Stir in the soy sauce and lime juice. Simmer for 2 more minutes to combine flavors.
Serve the curry over cooked rice and top with fresh cilantro if desired.