Alternatives to Jech for forcing via Boolean algebras As the title suggests, apart from Jech's Set Theory, are there any other noteworthy books or notes which cover forcing through Boolean algebras? All other texts I've seen use the poset approach (Kunen, Weaver, etc).

I'm going through the chapter on Jech's book, and having somewhere else to look for when I get stuck would probably be useful.
Try Rosser's book, Independence Proofs.

Also, once you understand that these methods involve semiregular topologies, look at the section on the relatively prime integer topology in Counterexamples in Topology by Steen and Seebach as well as the embedding of minimal Hausdorff topologies into semiregular topologies in Extensions and Absolutes of Hausdorff Spaces by Porter and Woods.  With regard to the latter, you will find that there are three kinds of extensions to H-closed spaces which need not be the same.  This will seem somewhat disconnected from set theory.  However, in so far as set theory presumably encompasses the "philosophy of the infinite", this three-fold multiplicity seems related to the fact that the free orthomodular lattice on countably many generators can be embedded into the free orthomodular lattice on three generators.

I had been led to these resources by my own curiosity and Jech's presentation of Boolean-valued forcing.  If formulating a complete Boolean algebra from a given separative partial order involves topologies formed from regular open sets, then one should be asking questions of what role semiregular topologies play in foundations.  One cannot do this without learning about instances of semiregular topologies elsewhere in mathematics.
Try Rosser's book, Independence Proofs.

Also, once you understand that these methods involve semiregular topologies, look at the section on the relatively prime integer topology in Counterexamples in Topology by Steen and Seebach as well as the embedding of minimal Hausdorff topologies into semiregular topologies in Extensions and Absolutes of Hausdorff Spaces by Porter and Woods.  With regard to the latter, you will find that there are three kinds of extensions to H-closed spaces which need not be the same.  This will seem somewhat disconnected from set theory.  However, in so far as set theory presumably encompasses the "philosophy of the infinite", this three-fold multiplicity seems related to the fact that the free orthomodular lattice on countably many generators can be embedded into the free orthomodular lattice on three generators.

I had been led to these resources by my own curiosity and Jech's presentation of Boolean-valued forcing.  If formulating a complete Boolean algebra from a given separative partial order involves topologies formed from regular open sets, then one should be asking questions of what role semiregular topologies play in foundations.  One cannot do this without learning about instances of semiregular topologies elsewhere in mathematics.