Step 1

Line a baking sheet with a single layer of paper towels. Finely grate zest from 1 large lemon into a medium bowl; cut lemon into wedges and set aside for serving. Finely chop 2 garlic cloves and add to bowl with lemon zest.

Step 2

Pour extra-virgin olive oil into a small saucepan to come ½" up sides; clip a deep-fry thermometer to sides of pan. Heat oil over medium until thermometer registers 225°–250°. Working in 2 batches and returning oil to 225° between batches, fry 1 cup (loosely packed) sage leaves (make sure you’re wearing an apron) until oil stops bubbling and sage is crisp but still fairly green, about 2 minutes per batch. Using a slotted spoon, transfer to prepared baking sheet. Repeat with ½ cup (loosely packed) rosemary leaves, then 10 sprigs thyme, returning oil to 225°–250° between batches and transferring to same baking sheet.

Step 3

Next, set a fine-mesh sieve over a medium Dutch oven. Drain one 3.5-oz.-jar capers (⅓ cup plus 1 Tbsp. packed) and pat dry. Fry, stirring occasionally, until they burst open and are crisp and darkened to a deep green (the oil will bubble steadily at first but should be bubbling less frequently by the time capers are ready), about 5 minutes. Remove from heat and strain through prepared sieve. Let oil cool. You should have 2–3 Tbsp.; if you have more than that, pour off excess and reserve for another use (like frying toast or making a vinaigrette). Transfer capers to baking sheet with herbs.

Step 4

Pluck leaves off 1 large bunch parsley until you have 2 (loosely packed) cups; discard stems (or save for another use). Coarsely chop parsley leaves. Chop fried thyme into pieces about the same size as the parsley and add both to bowl with garlic and zest, then add sage, rosemary, and capers. Season gremolata with kosher salt and gently toss to combine.

Step 5

Trim dark green tops off 2½ lb. small or medium leeks (6–8); discard. Slice white and pale green parts into ¼"-thick rounds and rinse in a large bowl of water, swishing around to loosen any dirt. Drain and repeat as needed. Wipe out bowl and return leeks to bowl. Slice 3 celery stalks on a slight diagonal crosswise ¼" thick, then thinly slice remaining 5 garlic cloves. Add both to bowl with leeks along with 1¼ tsp. Diamond Crystal or ¾ tsp. Morton kosher salt.

Step 6

Heat oil in reserved Dutch oven over medium. While the oil is heating up, slice 1¼ lb. small fingerling or Yukon Gold potatoes into ¼"-thick disks.

Step 7

Once oil is hot, add leek mixture and stir to coat with slotted spoon; reserve bowl. Cover with a lid and cook until vegetables are tender but not mushy, about 5 minutes. Transfer vegetables back to reserved bowl.

Step 8

Transfer potatoes to pot and add 1 quart vegetable stock and remaining 1 Tbsp. Diamond Crystal or 1¾ tsp. Morton kosher salt. Heat over medium-high until rapidly simmering but not boiling, partially cover, and cook until potatoes are tender but not falling apart, 15–20 minutes. Return leek mixture to pot and add remaining 1 quart vegetable stock and ½ cup heavy cream; cook until just starting to simmer again. Remove from heat; taste and season with more salt if needed.

Step 9

Ladle soup into bowls; spoon a generous amount of gremolata on top and stir in. Serve with reserved lemon wedges for squeezing over.