For the oil: Put the olive oil, chives and some salt and pepper in a blender and blend until smooth. Strain, taste and season, and then set aside for later use.
For the hollandaise: Place the egg yolks in a medium stainless steel bowl, set over (not in) a pot of simmering water. Whisk the yolks until pale yellow and fluffy. Slowly whisk in the clarified butter a few tablespoons at a time, until thickened. Add the lemon juice, salt, zest, black pepper and cayenne. Cover and keep in a warm spot until ready to use.
For the benedict: Heat the oil in a medium cast-iron pan set over medium-high heat until almost smoking. Sprinkle the salmon with salt and pepper and place top-side down in the hot oil. Cook until deep golden brown, 3 to 4 minutes. Flip, reduce the heat to medium and cook until golden brown on the second side and medium-well doneness, another 3 to 5 minutes. Remove to a plate to rest the salmon.
Line a plate with paper towels. Fill a medium saucepan halfway with cold water and add the vinegar. Bring to a simmer over medium heat. Crack each egg into separate ramekins. Stir the water, then carefully slip a few of the eggs into the water and cook until the whites are set and the yolk still runny, 3 to 4 minutes. Transfer to the paper towels and season with salt and pepper. Repeat for the remaining eggs.
While the eggs are cooking, toast the bagels. Use a fork to break the salmon into large flakes. Butter each bagel half and season with salt and pepper. Top each half with some salmon, a poached egg and several slices of avocado. Spoon some hollandaise sauce over the tops, then drizzle with some chive oil and garnish with chopped chives. Serve immediately.
This recipe makes about 2/3 cup of the chive oil; any leftover oil can be refrigerated for up to 3 days.