Step 1Trim the tops and tails of the radishes and quarter them lengthwise. In a large, nonreactive bowl, toss them with 1 tablespoon kosher salt to coat well. Add water to cover and several ice cubes. Set aside until the radishes have softened slightly but are still crisp, about 2 hours. Alternatively, transfer the salted radishes without the water and ice cubes to a Japanese pickle maker and screw down the top.
Step 2While the radishes are softening, toast the mustard seed and black peppercorns in a small saucepan over medium-high heat until they are fragrant, about 2 minutes. Add the vinegar, one-half cup water, the sugar and the remaining one-fourth teaspoon salt and simmer 3 to 5 minutes. Remove from the heat and cool until the pickling mixture is just warm to the touch.
Step 3Rinse the radishes well and pat them dry with a towel. Place them in a small, nonreactive container and pour the warm pickling mixture over them to just cover. Seal tightly and refrigerate at least 1 day to mature.