Whisk the honey, soy sauce, garlic, Dijon mustard, vinegar, paprika, and black pepper in a small bowl. In the second bowl, whisk the cornstarch with 2 tablespoons chicken broth.

Pat the kielbasa rounds dry. Heat a large lidded skillet over medium-high heat for 1 minute, then add 2 teaspoons olive oil.

Arrange the kielbasa cut-side down. Brown for 4 to 5 minutes, turning once, then transfer to a plate.

Add the remaining oil, potatoes, carrots, and onion. Cook for 2 minutes, keeping most potatoes cut-side down.

Pour in the remaining broth. Cover and cook for 7 minutes, until the potatoes are nearly tender.

Add the broccoli and cover for 2 minutes. Uncover and cook for 1 to 2 minutes, until the broccoli is crisp-tender and most broth has evaporated.

Return the kielbasa to the skillet and pour in the honey mixture. Toss and simmer for 1 minute.

Stir the cornstarch slurry again, then add it to the skillet. Toss for 30 to 60 seconds, until the glaze is glossy and coats the ingredients. Remove from heat and add parsley if using.