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Wrap the block of tofu in a few layers of paper towel, then set on a dinner plate or cutting board. Weigh it down with a heavy object like a skillet, pot, or large can while you prepare the curry.
Heat the oil in a Dutch oven or wide, high-sided skillet over medium heat until shimmering. Add the onion, ginger, and garlic. Cook, stirring often, until softened, about 5 minutes. Season with the salt and pepper and add the curry paste. Cook, stirring occasionally, until fragrant, about 2 minutes. Slowly whisk in the coconut milk, broth, and brown sugar. Bring the mixture to a simmer. Meanwhile, cut the tofu into 1/2-inch cubes.
Once simmering, add the tofu and broccoli. Simmer, stirring occasionally, until the liquid has thickened slightly, about 12 minutes. Remove from the heat and stir in the lime juice, tamari or soy sauce, and basil. Serve over rice if desired.