Cut broccoli florets into bite sized pieces. Peel thick skin off of the stems, and cut 1/2" thick. Keep the florets and stems separate.
Heat 2 tsp of the olive oil in a large skillet over medium high heat, until just smoking. Add broccoli stems and cook without stirring until browned on bottom, about 2 minutes. Stir and add florets. Stir until florets are beginning to brown, 2 more minutes.
Add water and cover. Cook for 2 minutes until broccoli turns bright green but is still crisp.
Remove lid and stir in chickpeas, garlic, anchovies and red pepper flakes. Cook until broccoli is just tender and chickpeas are heated through, 2 more minutes.
Remove skillet from heat. Stir in remaining olive oil, parsley and lemon juice and season to taste with salt and pepper. Serve immediately.