Reduce heat to medium and add butter to the pan. When the butter has melted, whisk in flour, pepper, and the remaining 1/2 tsp. salt. Cook, whisking constantly, for 2 minutes. Gradually whisk in milk and broth. Cook, whisking, until thickened and bubbly, 1 to 2 minutes. Gradually whisk in 3/4 cup cheese until melted.