Heat a skillet over medium-high. Add the garlic; cook, turning often, until the skin is dark brown in spots and the garlic is tender, 6 to 8 minutes. Let cool slightly, then peel and mince. Stir in the mayo and lemon juice. Season the aioli with salt; cover and chill.
Using a fork, mash the avocado with the onion, chile and 1 tbsp. lime juice. Season the guacamole with salt.
In a medium bowl, toss the cabbage with 1 tbsp. oil and the remaining 1 tsp. lime juice. Season the slaw with salt.
Add 1 inch of oil to a large skillet. Heat over medium-high until a deep-fry thermometer registers 350 degrees . Place 1 cup cake flour in a large bowl; whisk in 1 1/4 cups cold water for the tempura batter. Place the remaining 1/4 cup flour on a plate. Working in two batches, roll the asparagus in the flour, then dip in the batter. Deep-fry until golden brown, about 5 minutes per batch. Transfer to paper towels. Season with salt; cut in half crosswise.
Place 2 tortillas on each plate. Top with a spoonful of guacamole, then the asparagus and cabbage slaw. Drizzle with the aioli.