Pat chicken pieces dry with paper towels - this is crucial for browning. Heat olive oil in a large deep skillet over medium-high heat until shimmering. Season chicken with 1/2 teaspoon salt and the black pepper. Add to the pan in a single layer and cook undisturbed for 4-5 minutes until deeply golden on the bottom. Flip and cook 2-3 minutes more until just cooked through. Transfer to a plate; do not wipe the pan.

Reduce heat to medium. Add broccoli to the same pan and cook 2 minutes, stirring occasionally. Add garlic and cook 30 seconds until fragrant. Pour in chicken broth, scraping up any browned bits from the bottom of the pan with a wooden spoon. Bring to a simmer.

Add tortellini to the simmering broth, stirring to separate. Cover and cook for 4-5 minutes, stirring once halfway through, until tortellini are tender and broccoli is bright green and crisp-tender. Most of the broth should be absorbed; if too dry, add 1/4 cup more broth.

Reduce heat to low. Stir in heavy cream and remaining 1/4 teaspoon salt. Simmer gently for 2 minutes until slightly thickened. Remove from heat and stir in Parmesan and cold butter until melted and glossy. Return chicken and any accumulated juices to the pan, folding gently to coat everything in sauce. Taste and adjust seasoning.

Divide among shallow bowls. Top with additional Parmesan and crushed red pepper if desired. Serve immediately while the sauce is still loose and creamy.