Put a large pot of water on to boil.
Place a large, nonstick skillet on the stove over medium-high heat.
Add 1 tablespoon olive oil, onion, and garlic to the pan. Season with salt and pepper. Cook for 4-5 minutes, stirring occasionally. Add a cup of water to the pan, reduce the heat to medium, and cook for another 5 minutes or so, until the onions and garlic are soft.
Scrape the softened onions and garlic into a blender along with any remaining liquid. Puree. If it’s not moving nicely, add a splash more water. Taste and adjust seasoning.
Return to the stove and salt the water in the large pot. Add the pasta and stir. When it is three minutes away from being done, add the broccoli.
While the pasta and broccoli cook, put the remaining oil in the nonstick skillet. Set the heat back to medium-high and once the oil shimmers, add the chicken. Season with a pinch of salt and cook, stirring occasionally, until cooked through.
Drain the pasta and broccoli and return it to the pot (make sure to reserve some of the cooking liquid). Add the chicken and the garlic and onion puree. Stir to combine. Add a splash of cooking liquid if it is too thick. Taste and adjust with salt and lemon juice, as needed. Serve immediately.