Combine the mustard, egg yolks, Tabasco, and Worcestershire in a bowl. Whisk briskly until the mixture gets noticeably lighter in both color and texture. Whisk in the chili sauce until blended.
Add the oil a few drops at a time while whisking briskly. When the sauce begins to thicken, add the lemon juice, salt, and white pepper, and increase the rate of adding oil. Don't add oil faster than in a thin stream, and keep whisking.
When all the oil is incorporated, add the green onions and celery seed and stir until well distributed. You can make the sauce ahead to this point and refrigerate in a closed container.
When ready to serve, put about three tablespoons of the sauce into the bottom of a bowl. Add the crabmeat and pour half of the remaining sauce over it. With a rubber spatula, gently (so as not to break the lumps) toss the crabmeat to coat it generously with the sauce. Add more sauce as needed, but save some of the sauce.
Place a thick slice of tomato on each serving plate. Arrange avocado slices and lettuce around the tomato. Spoon the crabmeat atop the tomato. Serve with the extra sauce in a server. (Most people want more of it for the greens and the avocados.)