In a large pan, cook the bacon over medium heat until brown and crispy. Drain on a paper-towel lined plate. When cool, crumble the bacon into small pieces
Meanwhile, heat a large pot of water to boiling and cook the gnocchi according to package directions. The dumplings are usually done when they float the surface of the water. Drain and set aside.
Warm the milk and chicken broth in a small saucepan on low heat, or in a microwavable cup/bowl for 1 minute on high.
In a large saute pan, melt the butter over medium heat. When the butter is melted, add the flour, salt and pepper and whisk for about 1 minute, until a paste forms and the flour has a chance to cook a bit.
Slowly add in the warm milk/broth mixture and whisk until the sauce comes together. Cook for 3-5 minutes until the sauce thickens.
Add the gruyere and provolone cheeses and stir until the cheese melts.
Turn off the heat and stir in the cooked, crumbled bacon and chives.
Add the cooked gnocchi to the pan and toss well with the sauce.