Delay-tolerant networks (DTNs) are characterized by a possible absence of end-to-end communication routes at any instant. Still, connectivity can generally be established over time and space. The optimality of a temporal path (journey) in this context can be defined in several terms, including topological (e.g. shortest in hops) and temporal (e.g. fastest, foremost). The combinatorial problem of computing shortest, foremost, and fastest journeys given full knowledge of the network schedule was addressed a decade ago (Bui-Xuan et al., 2003). A recent line of research has focused on the distributed version of this problem, where foremost, shortest or fastest broadcast are performed without knowing the schedule beforehand. In this paper we show how to build fastest broadcast trees (i.e., trees that minimize the global duration of the broadcast, however late the departure is) in Time-Varying Graphs where intermittent edges are available periodically (it is known that the problem is infeasible in the general case even if various parameters of the graph are know). We address the general case where contacts between nodes can have arbitrary durations and thus fastest routes may consist of a mixture of continuous and discontinuous segments (a more complex scenario than when contacts are punctual and thus routes are only discontinuous). Using the abstraction of T-CLOCKS to compute the temporal distances, we solve the fastest broadcast problem by first learning, at the emitter, what is its time of minimum temporal eccentricity (i.e. the fastest time to reach all the other nodes), and second by building a foremost broadcast tree relative to this particular emission date.