Preheat oven to 375 degrees.
Line a 10-inch tart pan with a removable bottom with the pie crust, pressing the crust into the bottom and halfway up the sides. Cover the bottom of the tart crust with parchment paper, and fill with pie weights, or alternately, prick the crust several times with a fork.
Bake tart crust 15-20 minutes, until it starts to turn golden. Remove from oven and let cool a bit.
Reduce oven temperature to 350 degrees.
Sauté onions in olive oil. Add salt, and 1 tsp of fresh thyme plus the 2 sprigs of thyme when onions start to cook down. Cook onions until they are completely soft and caramelized. Remove thyme sprigs.
Place Montasio in the bowl of a food processor fitted with the chopping blade, and process until finely grated. Add remaining 1/2 tsp fresh thyme to cheese, and process until finely chopped.
Beat eggs with half and half in a small measuring cup and set aside.
Remove any pie weights and parchment paper from the cooled tart crust.
Spread grated cheese-thyme mixture evenly over the bottom of the tart crust.
Spread caramelized onions over the cheese, covering the entire crust.
Drizzle egg-cream mixture evenly over onions.
Return tart to the oven and bake for another 30 minutes, or golden brown and filling is set.
Serve warm or at room temperature.