Bring a medium pot of well-salted water to a boil. Cook broccoli florets and stems until bright green and just tender, about 2 minutes. Remove with a slotted spoon and let cool slightly. (Reserve water.) Roughly chop 1 cup cooked florets; set aside. In a blender, puree remaining florets, almonds, basil, garlic, and lemon juice. Add olive oil. Season with salt and pepper; set aside.
Return water to a boil and cook rice noodles according to package instructions. Reserve 1 cup cooking water, drain noodles, and return to pot. Toss with pesto, adding reserved pasta water until creamy. Top each bowl with chopped florets, small basil leaves, and remaining almonds. Season with pepper.