Heat 1 tablespoon of the oil in a 12-inch nonstick skillet over medium-high heat until just smoking.
Add the chicken pieces in a single layer and cook for 1 minute without stirring.
Stir the chicken and continue to cook until most, but not all, of the pink color has disappeared, and the chicken is lightly browned around the edges, 1 to 2 minutes longer.
Transfer the chicken to a clean bowl and set aside.
Add the remaining tablespoon of oil, onion, and 1/2 teaspoon salt to the skillet.
Return the skillet to medium-high heat and cook, stirring often, until the onion is softened, 2 to 5 minutes.
Stir in the minced garlic, oregano, and red pepper flakes, and cook until fragrant, about 30 seconds.
Add the ziti, 2 cups of the water, and the broth.
Bring to a boil over high heat and cook until the liquid is very thick and syrupy and almost completely absorbed, 12 to 15 minutes.
Add the broccoli and the remaining 3/4 cup water.
Cover, reduce the heat to medium, and cook until the broccoli turns bright green and is almost tender, 3 to 5 minutes.
Uncover and return the heat to high.
Stir in the cream, Asiago cheese, and reserved chicken with any accumulated juices and continue to simmer, uncovered, until the sauce is thickened and the chicken is cooked and heated through, 1 to 2 minutes.
Turn off the heat, stir in the lemon juice and season with salt and pepper to taste.