Shred the sharp cheddar using the smaller holes of a box grater. Set aside.
Trim the broccoli crowns and separate the florets from the stems. Cut the florets into approximately 1 to 1½-inch pieces.
Chop the broccoli stems into ¼ to ½-inch pieces.
Finely chop the yellow onion and combine it with the chopped broccoli stems. Finely grate or mince the garlic.
Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium heat for about 30 seconds.
Add the onion, broccoli stems, 1 teaspoon kosher salt, and ¼ teaspoon black pepper.
Cook for 5–7 minutes, stirring occasionally, until the onion becomes translucent and the broccoli stems are nearly tender.
Add the broccoli florets, remaining ¾ teaspoon salt, remaining ¼ teaspoon black pepper, ground mustard, and cayenne pepper, if using.
Cook for 3–5 minutes, stirring frequently, until the broccoli turns vibrant green.
Add the garlic and cook for approximately 30 seconds, stirring constantly.
Stir in the dried orzo and toast it for about 1 minute.
Pour in the broth and scrape the bottom of the pot to release any flavorful browned bits.
Increase the heat and bring the mixture to a boil.
Reduce the heat to maintain a gentle simmer. Cook for 6–8 minutes, or according to the orzo package instructions, stirring about once every minute to prevent sticking.
When the orzo is about 1 minute away from al dente and most of the broth has been absorbed, pour in the milk.
Simmer for approximately 1 additional minute, until the pasta is al dente but still saucy.
Remove the pot from the heat. Add the butter and shredded cheddar.
Stir until completely melted and the Broccoli Cheddar Orzo becomes creamy, glossy, and evenly coated.
Let the orzo rest for 5 minutes to allow the sauce to thicken before serving.