Wrap the block of tofu in 2 layers of paper towels and lay on a plate. Invert a second plate on top of the tofu and weigh down with a 28-ounce can. Leave for 20 minutes, then cut the tofu into 1/4- to 1/2-inch cubes.
Heat the dashi in a 4-quart saucepan set over medium-high heat. When the dashi reaches 100ºF on an instant-read thermometer, ladle 1 cup into a small bowl. Add both the dark and light miso, and whisk until smooth.
Bring the remaining dashi to a bare simmer, approximately 10 minutes. Add the miso mixture and whisk to combine. Return to a slight simmer, being careful not to boil the mixture. Add the tofu and scallions and cook until heated through, about 1 minute. Remove from the heat, ladle into soup bowls, and serve immediately.