Melt 1/4 cup butter in a heavy saucepan over low heat; whisk in flour and chicken soup base until smooth. Cook, whisking constantly, 1 minute. Gradually whisk in 1 1/2 cups milk; cook over medium heat, whisking constantly, 3 to 5 minutes or until mixture is thickened and bubbly.
Remove from heat, and whisk in Colby-Jack cheese, horseradish, and pepper, whisking until cheese is melted.