Prepare fennel by removing the stocks and root end, and slicing the bulb in half. Remove the core from each half with your knife. Peel off the outer layer, and thinly slice each half on a mandoline. (Reserve fennel fronds for garnish.)Place fennel in a bowl and add 2 Tbsp. lemon juice and 1/4 tsp. salt; toss to combine. Let sit for 20 minutes while you prepare the remainder of salad.
Heat 1 Tbsp. of the olive oil in a large skillet over medium. Add breadcrumbs and sesame seeds; cook for 3 to 5 minutes, stirring often, until golden and toasted. Transfer to a bowl.Add remaining 2 Tbsp. olive oil to the skillet and increase heat to medium-high. Add asparagus and cook until barely tender, about 3 minutes. (The asparagus should still be bright green with some bite to it.) Season with 1/4 tsp. each of salt and black pepper, and remove pan from heat.Once breadcrumbs are slightly cool, stir in Parmesan cheese.
Prepare dressing by combining all ingredients (except water) in a medium bowl; whisk to combine. Whisk in water in 1 Tablespoon increments until the dressing is smooth and creamy.
Assemble salad by arranging asparagus on a large serving platter. Scatter marinated fennel overtop, followed by avocado slices. Squeeze remaining 1 Tbsp. lemon juice over avocado slices, and season them with 1/4 tsp. salt. Scatter panko mixture overtop, and drizzle with dressing. Garnish salad with fennel fronds and a few grinds of black pepper.
Notes
This salad is best enjoyed fresh, however it can be refrigerated and enjoyed as leftovers for 2 days. The panko will lose its crisp, however the flavors will stay sharp.