Lightly toast the pistachio meats in a small skillet over medium heat until fragrant (about 3-5 minutes). Once done, transfer them to a food processor fitted with a standard S-blade. Add the garlic and pulse until the nuts and garlic clove are minced. Then add the lemon juice, vinegar, oil, mustard, 1/4 teaspoon salt (reserve 1/2 teaspoon), sugar, and pepper. Whir until combined. The dressing should appear a bit coarse, but without any big chunks of garlic or nuts. Transfer the vinaigrette to a large bowl, remove the S-blade, and wipe down the inside of the food processor with a damp paper towel.
Remove the bottom 3-4 inches of each asparagus spear and peel the carrots. Discard scraps. Fit your food processor with the slicer disc and run the asparagus and carrots through the processor (in batches) to yield small thin shavings. Transfer the shavings to the bowl with the vinaigrette.
Lastly, add the shaved parmesan and remaining 1/2 teaspoon kosher salt to the salad. Gently toss with your hands or a large spatula. Transfer the salad to a serving dish and enjoy!
TIP A: If you do not have a food processor, you can still prepare a tasty version of this salad using a carrot peeler. Instead of disc-like shavings, you'll create long shavings along the length of each asparagus spear and carrot. Simply grasp each asparagus spear at the tip and shave long strips along the edge of the spear and into a large bowl. You may need to start at the bottom half of the spear and move your way up to the tip to keep it from breaking. Once you reach the tip of the spear and have just a slender piece remaining drop it into the bowl with the other shavings (I sort-of like the visual effect of having the tender asparagus tips in the salad). Repeat with the remaining spears and carrots. Discard any large pieces of carrot that become too difficult to shave (e.g. the core).
For the dressing, you can prepare it in a blunder or simply mince the pistachios and garlic by hand and whisk them into the other vinaigrette ingredients.