Peel the potatoes and cut them lengthwise into 1/4-inch-wide sticks. Place in a bowl of cold water to prevent browning.
Heat the oil in a large, deep pot over medium heat until a deep-frying thermometer registers 325 degrees F.
Remove the potatoes to paper towels and dry well. Working in batches, fry the potatoes until cooked through and tender but not yet colored, 4 to 5 minutes. Be sure the potatoes have plenty of room and gently swish them around as they are frying. Drain on a paper towel-lined baking sheet.
Preheat the oven to 300 degrees F and increase the temperature of the cooking oil to 375 degrees F.
In a medium bowl, combine the flour, seafood boil seasoning, baking soda and a pinch of salt. Make a well in the center of the flour mixture and gently stir in the cold beer. Mix until just combined.
When the oil is at temperature, dip the monkfish fillets in the batter, coating the fish generously. To add the fish to the oil, carefully dip about one-third to one-half of a fillet into the oil and allow the batter to start puffing up before gently sliding it in. In batches, fry the fillets until golden brown and crispy, turning once, about 5 minutes. Drain on paper towels to blot the excess oil. Arrange the fish on a baking sheet lined with a cooling rack and keep warm in the oven until ready to serve.
Again working in batches, fry the potatoes a second time, at 375 degrees F, until golden brown and crispy. Drain on paper towels and immediately toss with kosher salt.
Transfer the fish to a platter and serve with the fries and remoulade.
Combine the mayonnaise, cornichons, mustard, malt vinegar, celery salt, cayenne and shallots in a bowl. Season with salt, if needed. Cover and refrigerate until ready to use.