In a large bowl, thoroughly wash the radishes in 3 to 4 changes of water. Drain. Trim any unsightly ends and tops and cut the radishes in half. In another large bowl, toss the radishes with the salt. Let stand for 20 minutes.
To make the seasoning paste, in a small bowl, combine the broth, chile pepper flakes, ginger, garlic, and sugar. Set aside.
Drain the radishes in a colander set over a bowl, reserving the brined juice. Rinse any excess salt off the radishes and let drain for another 15 minutes.
In a large bowl, combine the radishes with the seasoning paste and toss until evenly coated. Pack the radishes into a quart-size jar. Pour the reserved brining liquid into the bowl that was used to coat the radishes with the seasonings, and swirl the liquid around to capture any leftover seasonings. Pour into the jar, cover with a lid, and let sit at room temperature for 1 day. Refrigerate and consume within 1 week.
Place the mushrooms in a medium bowl. Pour boiling water over the mushrooms. Stir in the salt. Cover and let sit for 20 minutes. Squeeze the water out of the mushrooms and allow the broth to cool to room temperature or refrigerate before using. Discard the mushrooms or store them in the broth to further infuse their flavor. Refrigerate, covered, for up to a week.
From The Kimchi Cookbook by Lauryn Chun