Check the salmon fillets for grates, season with salt and pepper on all sides, then dredge in flour.
Heat the frying oil over high heat, then add the salmon fillets. Cook the salmon fillets for about 1-2 minutes per side (on all four sides). If your pan has a thin bottom you may need to reduce the heat to medium. The fillets are ready to be flipped when they don't stick to the pan anymore. Make sure that the pan and oil are well-heated.
Pour off the remaining oil, and add the butter to the pan. When it's melted, add finely chopped garlic and cook for 30 seconds, then whisk in the flour.
Add the broth, cream, milk, lemon juice, and mustard. Cook for a couple of minutes until the sauce is thickened.
Take the pan off the heat, add the chopped dill, and season the sauce with salt and pepper, to taste.
Place the salmon fillets in the sauce to warm them up, if necessary.
Enjoy!