We chill shaped butter so that we can lock it into a dough (that has also been chilled) of the same consistency, as for puff pastry or croissants.
For pâte brisée we dice the butter and chill until it is very cold so that the flour will coat the butter as it's mixed, before it has a chance to soften.
We use room-temperature butter in a number of cookies, such as TKOs, shortbread, and speculoos, when we want the butter to combine easily with the other ingredients. We also use it in pâte sucrée.
Before we cream butter for some cookie doughs or some tart doughs, we want the butter to be so soft and creamy that it forms soft peaks and has a consistency like mayonnaise. Typically we warm the mixer bowl with the butter in it, by holding the bowl over a burner, or using a blowtorch against the outside of the bowl, to encourage the softening.
We add melted butter to the lemon-poppy seed muffins, blueberry muffins, savarins, and crêpes. When we want a richer flavor, we melt butter and cook it to the point that it browns and has a nutty aroma and flavor; we then add it to the financiers and rhubarb tart.