How to Make It
Bring milk to a low simmer in a saucepan over medium. Gradually whisk in grits. Let come to a boil. Cover; reduce heat to medium-low. Cook, whisking occasionally, until very thick, about 12 minutes. Whisk in 2 ounces of the grated cheese and 2 teaspoons of the salt. Pour grits onto a 13- x 9-inch rimmed baking sheet or quarter sheet pan, and spread evenly. Press plastic wrap directly on surface. Chill until set, at least 6 hours or up to 3 days.
Combine basil, parsley, olive oil, and pine nuts in a food processor. Process until a paste forms, 15 seconds. Add remaining 1 ounce grated cheese and 3/4 teaspoon salt. Process until smooth, 15 seconds. Transfer to a bowl. Press plastic wrap directly on surface of pesto, and chill.
Cut chilled grits into 24 (2 1/4- x 2-inch) rectangles. Cut each rectangle in half diagonally to yield 48 triangles. Working in batches, dredge triangles in cornmeal, shaking off any excess. Arrange on a baking sheet lined with parchment paper. Chill at least 15 minutes or up to 4 hours.
Pour oil to a depth of 2 inches into a Dutch oven; heat over high to 360°F. Fry triangles, in batches, in hot oil until golden brown, 1 1/2 to 2 minutes. Transfer to a baking sheet lined with paper towels to drain. Serve warm topped with pesto (reserving remaining pesto for another use). Garnish with shaved cheese and small basil leaves.