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Boron does not provide both electrons - one electron is from boron and one is from fluorine in each of the three bonds in BF3. Boron is satisfied with only 6 valence electrons because since it only has 3 valence electrons, it can only form a maximum of three bonds, unless there is another atom that is willing to provide both electrons for the bond. This happens in the ion BF4-.
BF4- undergoes a coordinate covalent bond which means that 2 electrons come from one atom which in this case would be the extra fluorine atom. The reason why in BF3 boron is stable with an incomplete octet is because fluorine has such a high ionization energy that it is unlikely to exist with a positive formal charge.
The four exceptions are hydrogen, lithium, beryllium, and boron. These are the first 5 elements on the periodic table, so you they have only 5,4,3,2, or 1 electrons to start with. Therefore they cannot form more bonds than valence electrons.
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