How to Make It
Prepare the hollandaise: Place butter in a small saucepan over medium-low; cook until completely melted, about 5 minutes. Using a spoon, carefully skim solids off top of melted butter; discard solids. Slowly pour melted butter from pan into a bowl, discarding any remaining solids left in pan.
Pour water to a depth of 1 inch into bottom of a double boiler over medium; bring to a boil. Reduce heat, and simmer. Combine egg yolks and cold water in the top portion of double boiler. Cook, whisking constantly, until foamy and slightly thickened, 6 to 7 minutes. Gradually add 3/4 cup of the clarified butter, about 1 tablespoon at a time, whisking until incorporated after each addition and mixture is thickened.
Whisk in lime juice, chopped dill, and salt until blended. (Reserve remaining clarified butter for another use.) Keep hollandaise warm.
Prepare the eggs: See "4 Steps to Foolproof Poached Eggs," below.
Place 2 muffin halves, cut side up, on each of 4 plates. Divide smoked salmon among muffin halves. Using a slotted spoon, remove eggs from skillet, and gently place 1 egg on top of smoked salmon on each muffin half. Top each serving with 1 to 2 tablespoons hollandaise. Top evenly with salmon roe. Garnish with dill sprigs, and serve immediately.
4 Steps to Foolproof Poached Eggs
1. Add water to a large skillet, filling two-thirds full.
2. Bring to a boil over high; reduce heat to medium, and simmer. Stir in vinegar.
3. Break each egg into a custard cup; gently slip eggs into water in skillet.
4. Cook until desired degree of doneness (3 minutes for runny yolks, or 4 to 5 minutes for cooked yolks).