In a jar with a tight-fitting lid, combine olive oil, lemon juice, mustard, garlic, ½ teaspoon salt, and ¼ teaspoon pepper. Secure lid on jar and shake vigorously to combine.
Sprinkle steak with remaining ¾ teaspoon salt and ½ teaspoon pepper.
Heat a heavy skillet over medium-high until very hot. Add vegetable oil and tilt skillet to coat with oil. Add steak and sear, turning once, until browned and cooked to desired degree of doneness, about 1½ minutes per side. Transfer to a cutting board and let rest 5 minutes, then thinly slice across the grain.
In a large bowl toss spring mix with dressing to taste. Reserve remaining dressing for another use.
Lay out wraps and divide spring mix between them. Top with steak, blue cheese, and avocado.