Preheat the oven to 350F and grease a 10-inch springform pan with the pat of butter.
In a food processor (or in a large mixing bowl if you choose to do this by hand), combine the remaining butter and the 1 cup sugar and process until creamy. Add the eggs one by one, the ricotta, and the vanilla extract and mix between each addition.
In a medium mixing bowl, sift together the flour, salt, and baking powder. Add the flour mixture and orange zest to the egg mixture. Mix until just combined; the batter will be thick. Fold in half of the pistachios (reserve the other half for topping) and pour the batter into the prepared pan, levelling the surface with a spatula.
Arrange the apricots skin side down over the batter in a regular pattern. It's ok to crowd them a little; they will shrink as they bake. Sprinkle with the remaining tablespoon of sugar and the remaining pistachios.
Bake for 1 hour, until a knife inserted in the centre comes out clean. Transfer the pan to a cooling rack and let rest for 10 minutes. Run a knife around the pan to loosen the cake, unclasp the sides of the pan, and let cool completely before serving. The cake tastes best fresh out of the oven, but it will keep for 2 days at room temperature, wrapped tightly in foil.