Place the cut vegetables in a large colander in the sink or over a bowl. Sprinkle them with 1 teaspoon of the salt and massage for about a minute, then allow them to sit for at least 15 minutes. (Beets will stain other vegetables. If you'd like to maintain the separate colors of various vegetables, salt and let them sit in separate strainers.)
Add the vinegar, water, sugar and the remaining ½ teaspoon salt to a small pot and place it over medium heat. Stir until the sugar has completely dissolved and the liquid is steaming hot.
Rinse the vegetables under cold running water for a few seconds. Drain and squeeze out all extra water. Place in storage container(s)** along with any of the optional chiles or spices. Pour the hot pickling liquid into the container(s), making sure the liquid fully covers the vegetables.
Let the vegetables soak for at least 1 hour before using, or 2 hours for larger shapes. Store refrigerated, covered by the pickling liquid, in an airtight container for up to 1 month.
*Daikon and other radish varieties (and turnips, too!) smell funky when stored airtight for a while. The flavor of the vegetables should be unaffected, but it never hurts to nibble a piece to make sure before including them in a dish.
**A wide-mouth canning jar makes it easier to fish the pickles out of the brine.