Whisk the eggs with the milk and half-and-half until frothy. Whisk in the flour, starting a few tablespoons at a time to prevent lumps, them whisk in the sugar, salt, and cardamom.
Warm a griddle or a large heavy skillet over medium heat. Pour a thin film of oil onto the griddle and add a teaspoon or two of butter. Pour or spoon out the batter onto the hot griddle, where it should sizzle and hiss. A scant 2 tablespoons of batter will make a 3-inch pancake, about as large as you'll want to go with this batter. Because they spread so much, pour the batter out quickly, in one motion, then pause long enough for the previous pancake to firm, so that the cakes don't run into each other. Make as many cakes as you can fit without crowding. (Stir the batter up from the bottom each time you make a cake.)
Cook the pancakes for about 1 minute, until their top surface is covered with tiny bubbles, like most American pancakes, or until the batter has lost its gloss and turned a couple of shades darker near the outside edges, depending on whether you've used more or less flour, respectively. Flip the pancakes with a thin-bladed spatula and cook until the second side is golden brown, a minute or two longer. Repeat with the remaining batter, adding a bit more oil and butter to the griddle as needed. Serve immediately with butter and preserves or fruit.
Recipe courtesy A REAL AMERICAN BREAKFAST (Morrow, 2002) by Bill and Cheryl Alters Jamison