Melt butter in a large, deep skillet (or sauté pan) over medium heat. Add flour, cook for 30 seconds stirring constantly, then add garlic and sauté until flour and garlic are golden, about 30 seconds longer.
While whisking, slowly pour in heavy cream and milk. Cook stirring constantly until mixture simmers and thickens slightly.
Remove from heat, stir in Parmigiano-Reggiano. Return to low heat, stir and let cheese melt through.
Cover and keep warm over warm heat, stirring occasionally, until pasta is finished.