In a small bowl, whisk together the dry mustard, sugar, and 2 tablespoons warm water; let stand 5 minutes. In a small sauté pan over moderate heat, warm 2 tablespoons of the oil. Add the garlic and shallot and sauté, stirring occasionally, for 1 minute.
In a large bowl using a balloon whisk, whip the egg yolks for 1 minute to lighten them. Whisk in the lemon juice, Dijon mustard, salt, and cayenne pepper. Add the dry-mustard mixture, along with the garlic and shallots, and stir until blended. Gradually drizzle in the remaining 3/4 cup plus 2 tablespoons oil, whisking to combine. Cover and chill at least 20 minutes before serving. DO AHEAD: The mustard sauce can be prepared ahead and stored, covered in the refrigerator, up to 24 hours.
To Serve Serve the mustard sauce in small bowls or ramekins, alongside the stone crab claws.
Cooks' note: Turbinado sugar is raw sugar that has been steam-cleaned. The coarse crystals are blond in color, with a delicate molasses flavor. Demerara sugar, light brown sugar, or raw sugar can be substituted.