Cold eggs hold their shape better than room-temperature eggs when poaching. If using farm-fresh eggs that have not been refrigerated, chill them for about 1 hour before poaching, if desired.
Whisk together lemon juice, Dijon, egg yolks, and 3 tablespoons water in a heatproof bowl. Melt butter in a saucepan over medium-high. When it boils, reduce heat to medium; simmer until foamy. Continue cooking, stirring occasionally and scraping bottom of pan, until foam subsides, butter turns golden brown with a nutty aroma, and milk solids separate into brown specks that sink to bottom, 2 to 7 minutes. Remove from heat. Very slowly drizzle into lemon mixture, whisking constantly, then season with salt and pepper. Return to saucepan; set aside.
Bring 1 inch water and vinegar to a simmer in a nonstick skillet over medium-high. Meanwhile, mash avocado in a bowl; season with salt and pepper. Toast muffins and spread avocado onto split sides; top with crab. Gently crack eggs into simmering water. Simmer until whites are set but yolks are still runny, 3 to 4 minutes. Transfer to a paper towel-lined plate with a slotted spoon; let drain 30 seconds, then place atop crab. Warm Hollandaise over low, whisking constantly (do not let boil), until thickened slightly and heated through, 1 minute. Remove from heat, stir in tarragon, and spoon evenly over eggs. Serve immediately.