Chill the rib-eye in the freezer for 10 minutes to make it easier to slice.
Mince 1 fresh jalapeno pepper after removing the seeds and ribs (flavor not heat is important because the domestic elk has such a delicate flavor). Brush the corn tortillas with a little olive oil and place them directly on a hot grill, turn often. You want them crisp and with grill marks.
Mix your vinaigrette using a whisk in a small metal bowl. Slice the raw elk as thinly as possible, arrange on a large chilled platter after tossing the baby arugula in the dressing and placing in the middle of the platter. Sprinkle the minced jalapeno on the meat and drizzle with the good olive oil and the lemon juice. Season with salt and pepper and garnish with the cheese. Break up the grilled tortillas by hand and use the pieces to enjoy the elk.