Combine 1 tablespoon of the flour with the salt and pepper in a small dish and stir together. Place the chopped chicken pieces into a Ziploc bag and add the flour mixture. Seal the bag and shake/massage to coat the chicken.
Drizzle 1 tablespoon of the oil in a 5 QT Dutch oven or large pot and bring to medium-high heat. Add the chicken to the hot oil and cook, stirring occasionally, until cooked through and browned. Remove the chicken to a side plate and cover with aluminum foil to keep warm.
Lower the heat to medium. Drizzle the remaining 1 ½ teaspoons of oil into the pot and add the tablespoon of butter. When the butter is melted, add the garlic and cook for 30 seconds. Add the remaining 2 tablespoons of flour and whish with the butter and oil to form a paste. Add the milk, broth, and lemon juice and stir, scraping up any browned bits on the bottom of the pot.
Add the spaghetti to the pot and stir, covering it with the liquid. Increase the heat to medium-high and bring the mixture to a boil. Reduce the heat to low to simmer. Simmer for 10 minutes, stirring every couple minutes to prevent sticking, until the pasta is cooked through and has absorbed much of the liquid.
Stir in the rinsed capers, lemon zest, parsley, and the cooked chicken. Add the broccoli florets in a single layer on top of the pasta and cover with the pot lid. Keep the lid on for 2 minutes to allow the broccoli florets to steam. Remove the lid. The broccoli should be bright green and softened while remaining firm. Stir together and remove from heat.