Bring a large pot of water to a boil over high heat, then salt it generously.

Add the broccoli and asparagus and cook until crisp-tender, about 2 minutes (Resist the urge to cook more than that; the veggies continue to cook as they sit, and cook again in the pan with the sauce).

Transfer broccoli and asparagus with a slotted spoon to a bowl. Return the water to a boil, add the pasta and cook, stirring occasionally until al dente to tender but not mushy, about 8 minutes.

While the pasta cooks, combine the oil, red peppers and red pepper flakes in a large skillet. Cook over medium-high heat, stirring, until lightly browned and fragrant, about 2 minutes. Add the minced garlic and cook one more minute. Add the cherry tomatoes and cook, stirring, until somewhat soft, about 1 minute. Be careful not to cook any more, or the garlic will burn. Add 8.25 ounces of chicken broth, the remaining 2 teaspoons salt, and stir in 4 Tbsp butter and 1/4 cup of the Pecorino until incorporated. Stir back in the vegetables, bring to a simmer, and remove from the heat.

Drain the pasta in a colander in the sink. Transfer to a bowl and toss with the remaining 1/4 cup Pecorino. Pour the sauce and veggies over the pasta, season with pepper, and toss to combine. (If the sauce around the pasta seems thick, add a little more broth or water pasta cooked in, to give it a more velvety sauce. Serve in large bowls and pass additional cheese at the table, if desired.