For dressing, whisk mayonnaise, milk, mustard and garlic. Stir in lemon juice and Worschestershire (lemon juice will thicken the dressing).
Stir in Parmesan, season and chill until ready to use.
For fries, cut potatoes (peeled or unpeeled) into fries ½-inch (15 mm) in diameter and pat dry with paper towel. Heat oil in a deep pot to 350°F (180°C).
Add half the potatoes and cook until tender, about 5 minutes – the fries will not brown at this point. Remove fries with a slotted spoon onto a paper towel-lined tray and repeat with other half.
Fries at this point can be chilled until ready to cook and serve.
For second cooking, heat oil to 375°F (190°C). Add half the fries and cook until golden brown, about 5 minutes.
Remove from oil with a slotted spoon, let drain on paper towel for a moment, then toss in a bowl with a drizzle of Caesar dressing, salt and pepper.
Repeat with remaining fries and serve immediately.