Line a small rimmed baking sheet with parchment paper or plastic wrap, lightly coat the surface with olive oil, and set by the stove.
In a medium pot, add the olive oil, butter, kosher salt, and 4 cups cold water. Bring to a simmer over high heat, then whisk in the chickpea flour and lower the heat to maintain a strong simmer. Cook, whisking constantly, until the mixture thickens and starts to bubble, then switch to a wooden spoon and continue cooking, stirring continuously, until the mixture is very thick and smooth, 8–10 minutes. Scrape onto the prepared baking sheet, and use an offset spatula or the back of a spoon to smooth into an even ½-inch-thick rectangle. Cover tightly with plastic wrap and refrigerate until dense and set, at least 3 hours and up to 1 day.
When you are ready to fry, unwrap the rectangle and unmold it onto a large cutting board. Using a long, sharp knife, cut into batons, each about 3 inches long and ½ inch thick. Line a large heatproof plate or baking sheet with paper towels and set by the stove.
In a large skillet, add enough vegetable oil to come ¼ inch up the sides of the skillet. Heat over medium-high until shimmering, then add the batons in batches, taking care not to crowd the pan. Cook until golden and crisp on the bottom, 3–5 minutes. Using a slotted spoon, flip and continue cooking until deeply golden on the other side, 2–3 minutes more. Transfer the panisses to the prepared plate, and season generously with flaky salt and pepper. Repeat with the remaining batons, adding more vegetable oil as needed. Serve hot.