Cook bacon in a skillet over medium heat until crisp, about 10 minutes. Transfer to a paper-towel-lined plate to drain.
Melt butter in a large pot over medium-high heat until foamy. Add leeks and 1 teaspoon salt. Cook, stirring occasionally, until leeks are translucent, about 3 minutes. Add spinach, cover, and cook 2 minutes, stirring halfway through. Add peas and broth, and cook, stirring occasionally, until peas are tender and bright green, 3 to 4 minutes.
Stir in herbs and bacon, and season with salt and pepper. Divide lettuce leaves among 6 plates, and top with pea mixture. Serve immediately.