1. Cook first 5 ingredients in a small saucepan over medium heat, stirring occasionally, 8 to 10 minutes or until liquid has evaporated.
2. Pour water to depth of 1 1/2 inches into a 3 1/2-qt. saucepan over medium-high heat; bring to a boil. Reduce heat to medium, and simmer.
3. Whisk together egg yolks, salt, and red pepper in a 3-qt. heatproof bowl; gradually add lemon juice, whisking constantly. Place bowl over simmering water, and cook, whisking constantly, 3 minutes or until mixture thickens.
4. Whisk butter pieces, in 3 batches, into egg mixture, whisking constantly until butter is melted and mixture is smooth and thickened. Remove from heat. Stir in shallot mixture. Serve immediately.
Speedy Béarnaise Sauce: Prepare 1 (0.9-oz.) envelope béarnaise sauce mix according to package directions, stirring in 1 Tbsp. lemon juice and 1 Tbsp. chopped fresh tarragon during last minute of cook time.
Note: For testing purposes only, we used McCormick Béarnaise Sauce Mix. The classic Béarnaise Sauce cannot be made ahead. However, Speedy Béarnaise Sauce can be made 1 day ahead and stored in the refrigerator. To reheat, place bowl over simmering water, and cook, whisking constantly, until mixture is warm.