Put gastrique ingredients in a saucepan. Bring to a boil over medium-high heat. Reduce heat. Simmer for 3 to 6 minutes until reduced by half. Strain, cool, set aside.
For hollandaise, clarify butter by melting it in a saucepan. Skim foam from the top, then pour into a measuring cup, discarding any solids. Cover and keep warm.
Fill a large, wide pan or pot with 3 inches of water. Bring to a gentle simmer, then add vinegar.
Return to hollandaise. Fill a medium saucepan with 1 inch of water and bring to a simmer. In a medium bowl, whisk yolks and 2 tbsp cooled gastrique until frothy. Sprinkle with salt and pepper. Place bowl over the saucepan of barely simmering water (make sure it fits without touching the water). Whisk for 2 to 4 minutes until mixture is thick and frothy. Remove from heat. Place bowl on counter, wrapping base in a towel. Whisk in 1 cup warm clarified butter in a steady stream. Whisk in Tabasco, lemon juice and Worcestershire. Keep warm.
Cook ham for 1 to 2 minutes per side in a medium skillet on medium-high heat. Toast brioche until lightly golden. Remove to plates; layer ham over brioche.
Break 4 eggs into the large pot of simmering water. Cook for 2 to 3 minutes until whites are firm but yolks are still liquidy soft. You’ll need to cook eggs in batches. Remove eggs from water with a slotted spoon and set over a plate.
Place a poached egg over each ham-topped toasted brioche. Spoon hollandaise over top, then sprinkle with chives. Serve immediately.