Start by preparing the ingredients. Slice the grilled chicken fillet into equal, bite-sized pieces. Clean the fennel bulb, remove the tough core and shave or slice the vegetable paper-thin. With a sharp knife, cut away the peel of the orange down to the flesh and carefully cut the segments out between the membranes. Catch any juice that drips.
Place a dry skillet on the heat and let it get very hot. Toast the sliced almonds in it until golden. Keep shaking the pan regularly or stir with a spatula, as the nuts can burn quickly. Remove the almonds from the pan as soon as they're colored and let them cool on a cold plate.
Make a simple dressing by mixing a splash of olive oil with some apple cider vinegar and the reserved orange juice. Take a large bowl and add the sliced fennel, orange segments, chicken pieces and arugula. Pour the dressing over and toss everything well so all ingredients are coated with a light layer.
Divide the mixed salad among plates or arrange on a large flat platter. Sprinkle the toasted sliced almonds over the salad just before serving so this element doesn't soften from the dressing and retains its crispy bite.