Coat an 8½×12-inch baking sheet (quarter sheet pan) with nonstick spray.

Boil water in a large saucepan over high heat. Gradually add polenta, stirring constantly with a whisk. Reduce heat to medium; cook polenta until thick and beginning to pull away from sides of pan, 3–5 minutes.

Off heat, stir in ½ cup Parmesan and 2 tablespoons oil; season with salt and pepper. Spread polenta evenly onto prepared quarter baking sheet. Press plastic wrap onto surface; chill until firm, at least 1 hour or up to 2 days.

Heat oven to 400° with rack in center position. Coat a large baking sheet with nonstick spray.

Uncover polenta and turn polenta out onto a cutting board by placing a cutting board over sheet and flipping upside down; tap to release polenta from sheet. Cut polenta into 4×½-inch strips. Separate fries and arrange on prepared baking sheet in a single layer; drizzle with remaining 2 tablespoons oil and season with salt and pepper.

Bake fries 30 minutes, then flip and continue baking until brown, 10–20 minutes turning fries every 8 minutes or so to achieve even browning. 

Meanwhile, whisk together mayonnaise, remaining 2 tablespoons Parmesan, chives, 2 teaspoons lemon juice, ½ teaspoon lemon zest, and garlic; season with black pepper to taste. 

Let fries cool briefly on pan to crisp before serving with Parmesan aioli.