Cut off asparagus tips and set aside (they will be blanched and used as a garnish). Chop remaining spears into 1/2-inch pieces.: The delicate tips make a lovely garnish when blanched, offering a tender contrast to the blended body of the soup. As you trim, notice the clean snap of fresh asparagus and the bright green inside. Chopping the remaining spears into uniform 1/2 inch pieces ensures even cooking, and smaller pieces help the blender process everything smoothly. A common mistake is leaving uneven pieces which can cause some parts to overcook while others stay firm. If your stalks are uneven, shave the thicker parts with a peeler so cooking times match. Keep the reserved tips whole and set them aside on a plate while you continue.
In a small skillet or saucepan, bring 1/2 cup water to boil. Add asparagus tips, cover, and cook until tender, about 2 minutes. Remove from the pan and set aside to cool.: When the water hits a full boil, add the reserved tips and cover immediately so they steam rather than simmer aggressively. You want the tips to stay vivid green and tender, not mushy. After about 2 minutes they should be just tender when pierced with a fork, and their bright color will be locked in. Remove them quickly and let them cool on a plate so they do not continue to cook from residual heat. Overcooking here results in limp garnish and a dull color, which undermines the presentation. Use cold water or an ice bath if you want to stop the cooking instantly for extra snap.
In a large saucepan over medium heat, melt butter. Stir in chopped asparagus spears and leeks and cook until softened, about 8 to 10 minutes.: As the butter gently melts, it should foam slightly and carry the aroma of the leeks and asparagus. Add the chopped spears and sliced leeks and keep the heat at medium so they sweat and soften slowly. You want the vegetables to collapse and release their moisture without taking on brown spots. The sound will be a soft sizzle and the pan should smell sweet and green after several minutes. This slow softening builds flavor and prevents harsh raw onion notes. A typical mistake is turning the heat too high, which browns the vegetables and introduces bitterness. Stir every couple of minutes and scrape the pan bottom so nothing sticks.
Add broth and bring to a simmer. Cook until vegetables are completely tender, about 5 minutes. Stir in peas and Parmesan.: Pour in the broth and raise the heat to bring the pot to a gentle simmer, where small bubbles rise steadily but do not boil furiously. The broth will take on the green scent of the cooked vegetables. Simmering softens any remaining texture so the mixture blends silky later. The visual cue is that the spears and leeks will look translucent and will fall apart easily when nudged with a spoon. A common pitfall is undercooking the vegetables before blending, which results in a slightly fibrous or gritty mouthfeel. If you notice resistance when mashing a piece against the side of the pot, cook a few minutes longer.
Working in batches, transfer to a blender and process soup until smooth, about 2 to 3 minutes. Wash out saucepan, then add back blended soup.: Adding the peas and freshly grated Parmesan cheese at this point enriches the pot. The peas will cook quickly in the hot broth and boost the soup's verdant color, while the Parmesan melts into the liquid and deepens the savory profile. As the cheese warms, you will smell a warm, nutty note that signals the soup is gaining complexity. Stir well so the cheese dissolves and does not clump. Avoid adding pre grated dry cheese because it can be grainy; freshly grated melts more smoothly. If your soup seems bland after adding cheese, a pinch more salt will balance it.
Over medium-low heat, stir in half-and-half and lemon juice. Season to taste with salt (I like 3/4 teaspoon). Garnish with reserved asparagus tips.: Hot liquids expand in a blender, so work in batches and vent the lid slightly to release steam safely. As you blend, watch the texture change from chunky to glossy and listen for the blender's steady hum as pieces break down. The soup should be uniformly smooth with no fibrous bits remaining. If it seems coarse, blend a little longer. After pouring back into the cleaned saucepan, scrape the sides so every bit returns to the pot. A mistake here is overfilling the blender which can cause splatters. Use smaller batches for safety and consistency.
Over medium-low heat, stir in half-and-half and lemon juice. Season to taste with salt (I like 3/4 teaspoon). Garnish with reserved asparagus tips: Lowering to medium low prevents the dairy from curdling as you stir in the half and half and the bright hint of lemon juice lifts the flavors. Stir gently until the soup is warmed through and silky. Taste and season with salt gradually until the balance feels right, I often aim for about 3/4 teaspoon but adjust to your broth's saltiness. The final texture should coat the spoon and move in slow, glossy ribbons as you stir. Add the chilled blanched tips on top to finish the bowl with a tender bite and a pretty accent. Watch for boiling once the dairy is in; avoid boiling to keep the texture smooth and prevent separation.