Dissolve the tablespoon kosher salt in 2 cups water in a large bowl. Add the cabbage and soak for 30 minutes. Rinse the cabbage and drain well. Return it to the bowl and combine with the carrots and the scallions.
To make the dressing, whisk the eggs in a medium bowl until fluffy. Gradually whisk in the vinegar a little at a time. Add the sugar, mustard, salt, and pepper; whisk until uniformly combined. Transfer the mixture to a 2-quart saucepan and place it over medium-low heat. Heat slowly, stirring constantly, until the mixture reaches a gentle boil. Continue cooking and stirring until the mixture thickens, about 1 minute. Cool, then stir in the sour cream.
Add dressing to slaw and mix well. Refrigerate the slaw for 1 to 2 hours. Serve slightly chilled.