Put a steamer basket in a large pot and fill the pot with water to just reach the bottom of the basket.
Put the potatoes in a single layer in the steamer basket, set the pot over medium-high heat, and bring the water to a boil. Cover the pot and cook for 4 minutes, then carefully remove the lid, move the potatoes to one side of the pot, and add the carrots in a snug, slightly overlapping layer. Cover the pot and steam until the carrots and potatoes are just tender, another 6 to 7 minutes. Transfer the potatoes and carrots to a platter. Put the tofu, broccoli, and beans in the steamer; cover and cook until the tofu is hot and the broccoli and beans are just tender, about 4 minutes. Transfer to the platter with the other vegetables.
In a medium bowl, combine the peanut butter, soy sauce, chile paste, and 1/2 cup hot water from the pot. Whisk to combine, adding more water as needed to create a thick but fluid sauce. Add more chile paste and salt to taste. Serve with the sauce on the side.