Combine baby arugula, garlic, Asiago cheese, olive oil, and walnuts in a food processor and mix until smooth. Reserve ½ cup of pesto for later use.
Combine white wine and the rest of the prepared pesto in a sealed bag and carefully shake to mix. Add salmon filets and coat with the mixture. Seal and refrigerate for 1 hour.
Preheat grill-pan to medium-high. Oil the pan well to prevent sticking. Remove salmon from marinade and place on hot grill skin side up. Sprinkle with salt and pepper to taste. Grill 4–5 minutes per side. The fish is done when it's opaque all the way through. As you take the salmon off the grill pan, remove the skin and discard. Thin the reserved pesto sauce with a little olive oil and brush filets with the sauce. Enjoy!
If you’re pressed for time, no big deal! Just pick up a container of pesto at the store—it’s usually available in the deli section.