Heat the oven to 190 C / 375 F with a rack in the middle. Rub the butter over the inside of a 24 cm / 9.5-inch ceramic pie dish (about 5 cm / 2 inches deep), then tip in the 1 tablespoon sugar and tilt the dish until the surface is evenly coated.
Arrange the plums cut side down in a single snug layer in the dish. If they look very juicy, pat them dry with a paper towel first.
Blend the eggs, 80 g sugar, flour, cardamom, salt, milk, cream, vanilla, and orange zest in a blender for 20 seconds, until smooth with no flour streaks. A whisk and a 2-liter / 2-quart bowl works too. Pour the batter slowly around and over the plums.
Bake on the middle rack for 38 to 42 minutes, until the top is puffed and golden at the edges, the center barely jiggles when you nudge the dish, and a knife inserted between center and edge comes out clean. Pull it on time, because past about 45 minutes the custard turns rubbery and weeps.
Cool the clafoutis in the dish on a wire rack for 15 minutes. It will sink and wrinkle, which is normal. Dust with powdered sugar through a fine-mesh sieve, then cut into 6 wedges and serve warm.