In this joint work with Javier Jiménez-Garrido and Javier Sanz (Universidad de Valladolid) we introduce ultraholomorphic classes of functions in unbounded sectors of the Riemann surface of the logarithm defined by means of a weight function (of both Roumieu- and Beurling-type). For such classes we prove an extension theorem following the idea of the (single) weight sequence case from V. Thilliez. We are using real methods and applying the ultradifferentiable Whitney extension theorem for so-called Braun-Meise-Taylor classes (shown by Bonet/Braun/Meise/Taylor). The Roumieu case is obtained directly, the Beurling case follows by a reduction from the Roumieu case. The opening of the sectors for which the main result applies is controlled by above by a new growth index for weight functions (analogously as in the weight sequence situation treated by V. Thilliez).