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Gently stir water, egg yolks, and lemon juice in bowl until no streaks of yolk remain. Microwave, stirring gently every 10 seconds, until mixture thickens slightly and registers 160 to 165 degrees, 1 to 2 minutes. Immediately add ¼ cup oil, salt, mustard, and sugar; whisk to combine. (Tiny droplets of oil will float to top of mixture.)
Strain mixture through fine-mesh strainer into bowl of food processor. With processor running, slowly drizzle in remaining 1¼ cups oil in thin stream, about 2 minutes. Scrape bottom and sides of bowl and process 5 seconds longer. Transfer to airtight container and refrigerate for up to 1 month.