Preheat the oven to 180C/160C Fan/Gas 4. Grease and line the base of a 23cm/9in springform cake tin.
For the cake, melt the butter in a saucepan then whisk in the sugar, eggs and orange oil until smooth and well combined.
Sift the flour, bicarbonate of soda and ground spices into a large mixing bowl. Stir in the salt. Fold the butter mixture into the dry ingredients until combined. Fold in the carrots, sultanas, pecans, orange zest and juice. Pour the mixture into the prepared tin and bake for 1 hour, or until a skewer comes out clean when inserted into the middle (check after 45 minutes and cover the top of the cake with baking parchment if the cake is browning too much). Leave to cool in the tin for a few minutes then transfer to a wire rack to cool completely.
For the glaze, put the orange juice and sugar in a saucepan and cook until the volume of liquid has reduced to a sticky glaze. Brush the top of the cake with the glaze while it is still warm.
For the icing, slowly beat the butter and icing sugar together using an electric mixer. When combined, add the vanilla paste and orange oil. Continue to mix for another few minutes until light and fluffy. With the mixer on the slowest speed, add the cream cheese bit by bit until just incorporated.
Set aside a quarter of the icing and add a little orange food colouring, transfer to a piping bag fitted with a basket weave nozzle. Set aside a few tablespoons of the icing, mix in the green food colouring and transfer to a piping bag fitted with a small plain nozzle. (The coloured icings will be used to pipe a carrot shape on top of the cake.)
Cut the cooled cake in half horizontally and spread a third of the plain icing on the bottom layer, then replace the top. Neatly cover the top and sides of the cake with icing. Set aside to allow the icing form a crust. Transfer any remaining plain icing into a piping bag fitted with a small star nozzle.
For the decorations, pipe a carrot on top of the cake using the orange and green icings. Pipe stars of the plain icing around the bottom of the cake. Decorate the sides of the cake with the pecan halves, using a little extra icing to help them stick.