Heat the oil over medium heat in a 5-quart stockpot, add the onion and sauté until translucent. Add the milk, chicken broth, and butter and bring to a simmer. Gradually whisk in the cornmeal, avoiding any lumps. Reduce the heat to low and continue to cook for about 10 minutes, stirring occasionally.
Remove the pot from the heat and add the ham, herbs, salt, pepper, and cheese. Stir until the cheese is melted and the ingredients are evenly distributed.
Spread the cornmeal mixture on the surface of a rimmed 9'' x 13'' cookie sheet with a heat-resistant spatula. Cover the mixture with plastic wrap, put on oven mitts, then evenly press the mixture down, spreading it out to the edges. Refrigerate the covered cornmeal mixture until completely cooled.
Heat the oil in a Dutch oven to 375°F. (Or, preheat a deep fryer.) Flip the polenta out of the cookie sheet onto a large cutting board. Cut the polenta into 1/2'' strips and then cut each strip into 3'' pieces. Fry, in batches, until deep golden brown. (Adding too many at once can cool the oil down.) Drain the fries on paper towels; serve immediately.
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