In a large Dutch oven, melt butter over medium heat. Add onion and garlic; cook, stirring occasionally, until onion is translucent, about 5-7 minutes. Add flour, whisking constantly for 2 minutes. Gradually whisk in the broth; bring to a boil. Reduce heat to medium-low; whisk constantly until the sauce thickens, about 2 minutes. Remove from the heat; whisk in the cream cheese, thyme, rosemary, mustard, salt, and pepper until smooth. Stir in chicken and rice. Taste and season with additional salt and pepper, if desired. Transfer the mixture to the prepared baking dish.