Combine potatoes, 1 teaspoon salt, and 1 1/2 quarts cold water in large pot and bring to boil over high heat.
Reduce heat to medium and simmer until potatoes are barely tender, about 6 minutes.
Drain potatoes in colander, being careful not to break them.
Transfer potatoes to rimmed baking sheet coated with 1 tablespoon of oil.
Drizzle remaining tablespoon oil over potatoes and season with salt and pepper to taste and set aside.
Light grill and heat until grate is hot, about 5 minutes.
Generously sprinkle both sides of steaks with salt and pepper.
Cook, uncovered, over hottest part of grill until well browned on one side, about 3 minutes.
Turn steaks; grill until well browned on second side, 3 minutes.
Once browned, move steaks to cooler part of grill.
Continue grilling, uncovered, to desired doneness, 7 to 8 minutes more for rare (120 degrees on instant-read thermometer), 9 to 10 minutes for medium-rare on rare side (125 degrees), 11 to 12 minutes for medium-rare on medium side (130 degrees), or 12 to 13 minutes for medium (135 to 140 degrees). Remove steaks from grill and let rest for 5 minutes.
Once steaks are moved to cooler part of grill, place potatoes on hotter part.
Cook, turning once, until grill marks appear and potato slices are cooked through, 6 to 8 minutes.
While meat and potatoes are cooking on the grill, prepare sauce by beating butter with large fork in medium bowl until light and fluffy.
Add cheese, shallot, parsley, garlic, 1/4 teaspoon salt, and 1/8 teaspoon pepper; set aside.
To serve, place one steak and some of potatoes. Top each steak with 2 tablespoons butter mixture and serve immediately.
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Grilled Strip Steaks and Potatoes With Blue Cheese Butter (cont.)