Cut potatoes into 1-inch cubes and place them in a bowl. In a separate small bowl, whisk together 1 1/2 tablespoons olive oil, 1/2 teaspoon salt, 1/2 teaspoon lemon pepper seasoning, and 1/2 teaspoon dried thyme. Pour this mixture over the potatoes and toss until evenly coated. While the potatoes marinate, heat your grill to medium heat. I find that letting the potatoes sit in the seasoning for just a few minutes helps them absorb more flavor.

Transfer the seasoned potatoes from Step 1 to a grill basket, spreading them in a single layer for even cooking. Place the basket on the medium-heat grill and cook for about 15 minutes, shaking the basket occasionally to ensure all sides get contact with the heat. The potatoes should be mostly tender with slight resistance when pierced with a fork—they don't need to be completely soft yet since they'll continue cooking when the asparagus is added.

While the potatoes are grilling, trim the asparagus to 5-inch lengths, removing the woody ends. Place the trimmed asparagus in a bowl and add the minced garlic, remaining 1 1/2 tablespoons olive oil, 1/4 teaspoon salt, and 1/4 teaspoon lemon pepper seasoning. Toss gently until every spear is lightly coated. I like to mince the garlic fresh right before adding it so it's at its most aromatic.

After the potatoes have cooked for 15 minutes, add the seasoned asparagus from Step 3 to the same grill basket, nestling the spears among the potatoes. Continue grilling for about 10 minutes, shaking the basket occasionally, until the asparagus is tender and lightly charred and the potatoes are fully cooked through. The asparagus will be done when it has a slight char and bends easily when pierced.