For the tarragon aioli: 

1. Stir the ingredients together in a small bowl. Cover and chill until serving time.

For the tempura:  

1. Pour at least 3 inches of oil into a large sturdy pot suitable for deep-frying and broad enough for the asparagus spears.
2. Place the asparagus on a plate near your frying pot. Arrange a rack with paper towels under the rack for draining the asparagus. Also nearby, have a broad shallow dish with 3/4 cup of the flour, a large bowl for mixing the batter (which needs to be done at the last minute), and a long-handled “spider” strainer and tongs for retrieving the asparagus.
3. Just before you are ready to serve the tempura, heat the oil to 375 degrees.
4. While the oil heats, mix the batter, first whisking together the ice water and egg yolks. Mix them vigorously until frothy. Whisk in the remaining 1¼ cups of the flour, just until lightly blended. Don’t worry about a few streaks or lumps of flour. You don’t want to overwork the batter.
5. Dredge the asparagus in the shallow dish of flour, a few spears at a time, then dip in the batter and slip into the oil one at a time. Repeat with remaining asparagus and batter, adding as many as you can to the oil without crowding or dropping the oil temperature. Cook the asparagus for 3 to 4 minutes, turning as needed, until golden.
6. Serve right away with the aioli and lemons to squeeze over the tempura.