Preheat the oven to 425 degrees F.
For the soup: In a large soup pot or Dutch oven, bring the oil to medium heat. Add the shallot and salt and pepper to taste. Sweat until translucent, about 1 minute, being careful not to brown. Add the asparagus to the pot and sweat until beginning to soften, about 3 minutes. Add the peas and sweat another minute. Pour in the white wine and deglaze the pot. Add 5 cups water and continue simmering until crisp-tender, another 10 minutes. Remove from the heat and set aside.
Working in batches (and carefully as the liquid will be hot), puree the soup until smooth in a blender. In the last batch, add the baby spinach and puree; this will give the soup a vibrant green color. Strain back into the pot through a fine mesh sieve. Return to a simmer until slightly thickened, 4 to 6 minutes. Season with salt and pepper. Remove from the heat.
For the Parmesan breadcrumbs: While the soup simmers, pulse the bread in a food processor until you have very coarse crumbs. Once processed, transfer the breadcrumbs to a medium mixing bowl. Drizzle with a generous amount of olive oil. Toss with the thyme. Season mixture with salt and pepper. Transfer to a parchment-lined sheet tray. Using a grater, finely grate a few tablespoons of Parmesan over the breadcrumbs. Place into the oven and bake until golden and crispy, 7 to 10 minutes. Remove from the oven and cool.
To serve: Ladle the soup into shallow serving bowls. Sprinkle evenly with a few teaspoons of Parmesan breadcrumbs and top with a dollop of creme fraiche. Garnish with chopped chives, cilantro leaves, a squeeze of a lime wedge and a splash of extra-virgin olive oil. Serve immediately.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.