In a small saucepan, melt the butter over medium-low heat. Ensure it melts completely without browning. Sprinkle the flour over the melted butter. Immediately begin to whisk the flour and butter together until they form a smooth, thick paste. This mixture is called a "roux." Continue to cook, whisking constantly, for about 1-2 minutes. This crucial step cooks out the raw taste of the flour and will give your sauce a clean, creamy flavor instead of a pasty one. The roux should be bubbly and smell slightly nutty.
Remove the saucepan from the heat for a moment. This helps prevent lumps, especially for beginners. Slowly pour in about a quarter of the milk while whisking vigorously. The mixture will seize up and become very thick at first—this is normal. Once it's smooth, continue to gradually add the remaining milk in a slow, steady stream, whisking constantly until all the milk is incorporated and the sauce is completely smooth. Make sure to whisk into the corners of the pan.
Return the saucepan to medium heat. Bring the sauce to a gentle simmer, stirring frequently. You should see small bubbles breaking the surface. Let it simmer for another 2-3 minutes, stirring, until the sauce has thickened enough to coat the back of a spoon. If you draw a line through the sauce on the spoon with your finger, it should hold its shape. The sauce will continue to thicken slightly as it cools.
Remove the sauce from the heat and stir in the salt and white pepper. Taste and adjust the seasoning if necessary. Use your beautiful Béchamel sauce immediately in your desired recipe.