Step 1

Combine shallots and ½ cup oil in a medium pot and set over medium heat. Cook, stirring occasionally, until shallots turn pale golden, 7–9 minutes. Add peanuts and turmeric and cook, stirring occasionally, until shallots are golden brown and mixture smells very nutty, about 2 minutes. Remove pot from heat and strain through a fine-mesh sieve set over a medium bowl. Let oil cool, then stir shallot mixture back in. Season with salt; set aside. Clean pot and reserve.

Step 2

Trim 1" from bottom of broccoli stalks, then peel. Remove stalks from crowns and thinly slice; set aside. Coarsely chop florets into 1"–2" pieces; set aside separately.

Step 3

Heat remaining 3 Tbsp. oil in reserved pot over medium-high. Add onion and cook, stirring occasionally, until softened and beginning to brown in spots, 7–9 minutes. Add chile and garlic and season with salt. Cook, stirring, until garlic softens, about 3 minutes. Add potato and reserved broccoli stalks to pot and stir to coat in onion mixture. Pour in 5 cups water, season with salt, and bring to a boil over high heat. Reduce heat to medium-low, cover pot, and simmer until potato and broccoli stalks are tender, 12–15 minutes.

Step 4

Add two thirds of reserved broccoli florets to pot, cover, and cook until bright green, about 4 minutes. Add spinach and cook, stirring, until wilted, about 1 minute. Remove pot from heat. Working in small batches, purée in a blender until very smooth. Transfer to a large bowl as you go, then return to pot. (Or use an immersion blender if you have one and blend directly in the pot.)

Step 5

Set pot over medium heat and add remaining broccoli florets. Cover pot and cook until florets are bright green, about 4 minutes. Stir in fish sauce; taste and season with more salt if needed.

Step 6

Divide soup among bowls and top with herbs and reserved shallot mixture.