Both main trailheads are possibles: Linza and Belagua. The shortest and most popular is the beginning in Refuge de Linza (and it's my recomendation).
The approach to both trailheads is in the main page.
The collectors of summits had two secondary peaks in this ridge, the Budogia Occidental (2345m), or Pico Linza, and the Budogia Oriental or main summit of Budogia (2367m).
The trailhead of this route is possible from Linza or Belagua.
In the route of Belagua go to south from the Col of Larreria in direction to Ukerdi. In the route of Linza go to west from the Col of Linza bordering the Budogia behind of this peak. Both routes bring to Portillo de Larra (3h) in the bottom of the ridge of Budogia.
Between the peaks Ukerdi and Budogia we'll follow the channel to the ridge of Budogia across a hard slope. The area is confuse without path and signals but the best option is walk near of the edge in right side. In some points we'll descnet a few of meters to evitate the edge but the direction is always evident. In a little col appear the cairns with the indication to turn to south (right). We 'll follow the cairns of stones and reach the first peak of Budogia, the Budogia Occidental (2345m), also called Pico Linza in some books and maps. The best option is to walk now in the spectacullar edge with impressive views to reach the summit of peak Budogia (2367m, 4h). A step-crossing of descent of grade I+ (or II-) bring to Col West of La Mesa. The ridge is narrow and vertical and it's not for hikers, it's the only point complicated of the ridge. It's very easy for mountaineers with experience in rock-climb but is impressive and vertical for the begginers. From the col we'll walk to the right to search the final chimney of grade I to the summit (6h from Belagua, 4h 30min from Linza).
Alternatives: if you want to climb only La Mesa and not the Budogia you can to search a walk under Budogia in an area of karst to reach the Col West of La Mesa between Budogia and La Mesa (point of entry as well of the normal route). I think this option had interest if you trailhead is Belagua but from Linza it's better the normal route if you don't want to climb Budogia.