So, in the early modern middle east and northern India, three states (the Ottomans, the Safavids, and the Mughals) were able to quickly adopt and establish local monopolies over gunpowder, and as a result were able to become vast space filling empires.
Your mission, should you choose to accept it, is to come up with a scenario where a similar circumstance allows one or more European country to achieve a similar degree of rapid expansion (and no, conquering a fairly empty region like Siberia doesn't count).