Follow these detailed steps carefully to ensure your polenta is perfectly creamy, sets up firmly, and fries to a crispy, golden-brown perfection. The process is straightforward but requires a bit of attention at the beginning.

Part 1: Cooking the Creamy Polenta Base

1. Prepare Your Pan: Lightly grease a 9×13 inch baking dish or a similar-sized pan with butter or a neutral oil. This will prevent the polenta from sticking and make it easy to remove later. Set it aside.
2. Boil the Liquids: In a large, heavy-bottomed pot or Dutch oven, combine the 4 cups of beef broth and 1 cup of milk. Bring the mixture to a rolling boil over medium-high heat.
3. Whisk in the Cornmeal: Once the liquid is boiling, reduce the heat to low. While whisking constantly and vigorously, slowly pour the 1 ½ cups of cornmeal into the pot in a thin, steady stream. Pouring it slowly while whisking is the most important step to prevent lumps from forming.
4. Cook and Stir: Continue to cook on low heat, switching from a whisk to a wooden spoon or heatproof spatula. Stir frequently, scraping the bottom and sides of the pot to prevent scorching. The polenta will begin to thicken significantly within the first 5-10 minutes. Cook for a total of 25-30 minutes. You’ll know it’s done when the polenta is very thick, pulls away from the sides of the pot as you stir, and the individual grains are tender (not gritty).
5. Add the Flavor: Remove the pot from the heat. Immediately stir in the 4 tablespoons of butter until it has completely melted. Next, add the ¾ cup of Parmesan cheese, ½ cup of cheddar cheese, the diced ham, fresh thyme, 1 teaspoon of salt, and ½ teaspoon of black pepper. Stir vigorously until the cheese is fully melted and all the ingredients are evenly distributed throughout the thick polenta.

Part 2: Chilling and Cutting

1. Spread and Chill: Immediately pour the hot polenta mixture into your prepared 9×13 inch pan. Use your spatula to spread it into an even, smooth layer.
2. Cool to Room Temperature: Let the polenta cool on the counter for about 1 hour, or until it’s no longer steaming.
3. Refrigerate Until Firm: Cover the pan with plastic wrap, pressing it directly onto the surface of the polenta to prevent a skin from forming. Refrigerate for at least 3 hours, or preferably overnight. The polenta must be completely cold and very firm to be cut cleanly. This step is not optional!

Part 3: Frying for a Crispy Finish

1. Cut the Bites: Once fully chilled, run a knife around the edge of the pan and carefully invert the firm polenta block onto a large cutting board. Cut the block into your desired shapes. Small squares or rectangles of about 1.5 to 2 inches work perfectly for bites.
2. Heat the Oil: In a large, heavy-bottomed skillet (cast iron works wonderfully), heat 3-4 tablespoons of olive or avocado oil over medium-high heat. The oil is ready when it shimmers and a small piece of polenta sizzles immediately upon contact.
3. Fry in Batches: Carefully place the polenta bites in a single layer in the hot skillet. Do not overcrowd the pan, as this will lower the oil temperature and result in soggy, greasy bites. Work in batches if necessary.
4. Cook Until Golden and Crispy: Fry the bites for 4-6 minutes per side, without moving them too much. This allows them to form a deep golden-brown, crispy crust. Flip them carefully with a thin spatula and cook for another 4-6 minutes on the other side.
5. Drain and Serve: Transfer the crispy polenta bites to a wire rack or a plate lined with paper towels to drain any excess oil. Serve immediately while hot and crispy.