Grilled Ribbons of Thai-Style Steak

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1 flank steak (about 2 pounds), very cold (See Cook's Note.)

1/2 cup Thai Red Curry Paste, recipe follows

3/4 cup unsweetened coconut milk

1/4 cup firmly packed dark brown sugar

1 tablespoon Southeast Asian fish sauce

1 teaspoon ground turmeric (optional)

Spray oil

Peanut Sauce, recipe follows

Thai Red Curry Paste

8 dried red chiles, like arbol, stemmed and seeded

1 tablespoon coriander seeds

2 teaspoons cumin seeds

1 teaspoon white peppercorns

1/4 cup packed roughly chopped coriander leaves, stems, and roots (cilantro)

2 tablespoons Southeast Asian fish sauce

2 teaspoons kosher salt

1/2 teaspoon ground cardamom (optional)

1/4 teaspoon ground nutmeg

2 medium fresh red chiles, stemmed and chopped

5 cloves garlic

3 shallots

2 fresh lemongrass stalks (about 6 ounces), trimmed and thinly sliced

1-inch piece fresh ginger, peeled and chopped

2 tablespoons vegetable oil, like soy, peanut, or corn

Peanut Sauce

2 tablespoons coconut cream skimmed from the top of canned coconut milk

1 tablespoon plus 1 teaspoon Thai Red Curry Paste, see above

1 cup chicken broth, homemade or low-sodium canned

1/4 cup unsweetened coconut milk

1/4 cup chunky peanut butter

1 tablespoon Southeast Asian fish sauce

1 tablespoon freshly squeezed lime juice, or more to taste