In a blender, combine the olive oil and the oregano and blend until smooth. Season lightly with salt and pepper. Pour oil in a bowl and set aside.
Put the avocados, garlic, serrano, yogurt, lemon zest and juice in the bowl of the food processor and process until smooth. If mixture is too thick or chunky, add a touch more yogurt and.or lemon juice so mixture processes thoroughly. Transfer the mixture to a bowl.
Add the cucumbers and parsley and season with salt and pepper. Stir to combine, cover and refrigerate for at least 30 minutes and up to 2 hours before serving.
When you are ready, fire up the grill and get it nice and hot. Brush both sides of the salmon filets with the oregano oil and season with salt and pepper. Grill the salmon for about 3-5 minutes per side, depending on the thickness. You want it to char and cook to medium. Brush with salmon with more oil while cooking.
While the fish is cooking, brush the cut side of the lemons with the reserved 2 tablespoons of olive oil and grill, cut side down, until lightly golden brown, about 2 minutes.
Serve the filets with a generous dollop of avocado-tzatziki and squeeze the juice from the grilled lemons on top.