Warm oil in a large pot over medium heat. Add onion; cook, stirring occasionally, until softened, 6 to 8 minutes. Squeeze spinach to extract any excess moisture. Raise heat to medium-high and stir spinach into pot, sprinkle flour over the top and mix well. Pour in milk and broth. Cook, stirring constantly (so mixture doesn't boil), until slightly thickened, about 8 minutes. Season with salt and pepper. Remove from heat and let cool slightly.
Working in batches, puree spinach mixture in a blender until smooth. Transfer all but 2 cups of soup to a large bowl. Add avocado to blender and puree until smooth. Stir into soup.
Cover and refrigerate until thoroughly chilled, at least 2 hours. Just before serving, stir in Tabasco and lemon juice. Season with additional salt and pepper, if desired. Serve chilled, garnished with lemon wedges, if desired.