In a small saucepan over medium-high heat, bring milk and 1 1/2 cups water to a simmer. While whisking, slowly pour grits into saucepan, then reduce heat to medium-low. Cook grits, uncovered and whisking frequently, until thickened, about 7 minutes.
Stir in cheese, 2 tablespoons butter, and 1 teaspoon Cajun seasoning; season with salt. Pour grits mixture into an 11" x 8" baking dish; spread in an even layer. Refrigerate until cold, about 30 minutes.
Meanwhile, pat shrimp dry and transfer to a large bowl; season with salt and 1 1/2 teaspoons Cajun seasoning, tossing to coat.
In a large skillet over medium heat, melt remaining 2 tablespoons butter. Add celery and bell peppers, season with salt. Cook, stirring occasionally, until peppers start to soften, 3 to 4 minutes. Add white scallion parts, garlic, and tomato paste and cook, stirring, until paste begins to darken, about 2 minutes.
Make a space in center of skillet and add seasoned shrimp. Cook, tossing occasionally, until shrimp are cooked through and opaque, 3 to 4 minutes. Transfer shrimp mixture to a plate. Wipe skillet clean.
Using 2" round cutter, cut out 15 cakes (they should be about 1" thick) from grits in baking dish and transfer to a baking sheet. Cut each cake in half through the equator (like a hamburger bun) to make 30 (1/2"-thick) disks.
In one shallow bowl, place flour. In a second shallow bowl, beat eggs to blend. In a third shallow bowl, place panko. Whisk remaining 1 teaspoon Cajun seasoning into flour. Working in batches, coat cakes in flour, then dip into egg, shaking off excess. Dip into panko, lightly pressing to adhere. Return breaded grits to baking sheet and freeze until cold, about 10 minutes.
In same skillet over medium-high heat, heat oil. Working in batches, fry cakes, turning halfway through, until golden brown and crisp, 1 to 2 minutes per side. Transfer to a paper towel-lined plate; season with a pinch of salt.
Transfer cakes to a platter. Top with a shrimp and some vegetable mixture. Sprinkle with green scallion parts. Serve with remoulade alongside (if using).