In a food processor or blender combine the fresh basil, almonds, Grana Padano or Parmesan cheese, pressed garlic clove and the juice of one lemon. Process until smooth. With the motor of the food processor still running pour in the olive oil through a chute and process for 10-20 seconds longer. Turn the food processor off and taste the pesto, add a pinch of salt if necessary.
Cook the pasta in a large pot of salted water according to package directions.
While the pasta is cooking, preheat the grill and grill the chicken breasts seasoned with salt, pepper and dried basil for 5-7 minutes per side, depending on the thickness of the breast. Remove to a separate plate and keep warm.
A couple minute before the pasta is fully cooked, heat 1 tbsp of olive oil in a large pan, then saute sliced garlic cloves over low heat for 10-20 seconds. Do not allow the garlic to brown, then add cherry tomatoes and arugula and cook while stirring for 1-2 minutes.
Drain the pasta while reserving 1 cup of cooking water. Add the pasta and pesto directly to the pan with garlic, cherry tomatoes and arugula, toss everything gently to combine, add a splash of reserved water to loosen the sauce. The dish should be saucy and not pasty. Keep adding the pasta water until you reach the desired consistency. Taste and add more salt if needed. Slice the grilled chicken and top the pasta with it. Serve with additional grated Grano Padano or Parmesan and a handful of fresh cherry tomatoes for extra texture and freshness.