It's a cross between pudding and cake - a new take on a traditional French dessert that is a perfect summer dessert.
Mix the Amaretto or other liqueur and 1/4 cup of the sugar. Then add the pitted fresh cherries. Let the cherries macerate in the liqueur and sugar mixture for at least 30 minutes, preferably one hour. Toss the cherries occasionally, so that the liquid gets inside all of them. (Macerating is the fruit equivalent of marinating, a term generally reserved for vegetables or meat.)
Preheat the oven to 425 degrees F.
Stir the flour and slat together in a large bowl. Whisk in about half of the milk, until the mixture is thin and smooth.
Add the eggs one at a time, whisking after each one. Then add the sugar and vanilla. Finally, add in the remaining milk and whisk until the batter is smooth. At the beginning, the batter is quite thick, but by the end it is more liquidy than pancake batter.
Melt one tablespoon of the butter in the cast iron pan. Move the butter all around so it greases the entire bottom and on the sides of the pan. Pour about 1/2 cup of the batter into the pan on a medium-high heat and tilt the pan or spread the batter with an implement. The batter should completely cover the bottom of the pan, like a crepe. Don't worry if a few bubbles appear. This step simply creates a skin onto which you put the rest of the clafoutis, and makes it easier to lift pieces out of the pan once it is done. (That technique is from Julia Child's cherry clafoutis in Mastering the Art of French Cooking.) Take the pan off the heat.
Drain the juice from the marinating cherries (reserve it for the optional cherry syrup), and place the cherries in the pan so that they do not overlap. There is no need to be obsessive - this dessert is rustic looking - but try to leave space between them. Gently pour in the rest of the batter (the cherries begin to float, so being gentle in how you pour keeps them from moving too much), then dot the top with the rest of the butter (1 & 1/2 tablespoons) and sprinkle on the sliced almonds.
To avoid drips in the oven, place a cookie sheet on the middle rack and put the pan on that sheet. Bake the clafoutis for 20-25 minutes until it is lightly browned. The USDA recommends making sure the internal temperature of the clafoutis (on an instant read thermometer) is between 160-165 degrees.
When it is fully baked, let the clafoutis cool in the pan on a wire rack until you can comfortably touch the pan handle without hot pads. Refrigerate the clafoutis and dust it generously with powdered sugar immediately before serving.
Put the pits and 1/4 cup of sugar in a small sealed jar or other glass container with the lime. Shake those ingredients to mix them. Let them macerate for 3-4 hours, shaking every once in a while, as you make and cool the clafoutis. Just before serving the clafoutis, strain the pits out and add as much of the reserved Amaretto/sugar mixture as you would like. Serve on the side.
If you don't have a 10-inch cast iron pan, another oven-proof pan, i.e. no wood or plastic works too. Another alternative is a deep dish metal pie pan. You cannot use a glass pie pan and do the "crepe skin" technique at the beginning of putting the clafoutis in the pan. However, in a pinch, you could make that as a 9-inch crepe in a pan on the stove (cooking only one side) and transfer it to a deep dish glass pie pan, proceeding from there.