Let cast-iron skillet cool 1 to 2 minutes, then return skillet to heat and reduce heat to low. Add ¼ cup unsalted butter.
Once butter has melted, add 2-3 cloves garlic to skillet and stir to incorporate minced garlic into butter. Sauté garlic 30 to 60 seconds or until just fragrant.
Once garlic is fragrant, add ½ cup low-sodium chicken broth and ¼ cup dry white wine to skillet. Stir to incorporate liquids, scraping bottom of skillet to loosen any bits of chicken or garlic that may have stuck to pan, then let liquids simmer over low heat until reduced by approximately 50%.
After reducing liquid, move skillet to cool burner and let skillet cool slightly, 30 to 60 seconds. When skillet has cooled, slowly add 2 cups heavy cream and gently whisk heavy cream into chicken broth reduction.
When heavy cream is incorporated, add 1 cup grated fresh parmesan to mixture in small increments. Whisk gently after each addition to melt parmesan and incorporate cheese into liquid.
Once all parmesan has been incorporated, return skillet to low heat. Gently whisk mixture 1 minute or so, until mixture is heated through.
Add ⅔ cup sun-dried tomatoes, 1 teaspoon dried oregano, and ½ teaspoon crushed red pepper flakes to mixture. Gently stir ingredients into cream sauce, then taste sauce and adjust salt, black pepper, oregano, and/or red pepper flakes as desired. Note: if sauce is thinner than desired, stir in 1 tablespoon of reserved pasta water then simmer sauce 1 to 2 minutes. Repeat if needed, but be careful not to add too much water.