It seems that the CpG islands seen here in the UCSC genome browser, are in the transcribed genomic region of the gene. If you drag the cursor to the left you may see the promoter region upstream and there may be other CpG islands there that may be important for gene expression. For the FASN gene in mouse, as you mention, it seems to be that one that overlaps with the promoter.
I have other question but it looks like stupid but I am starting with this kind of analysis: How can I say that the CpG Island is in the promoter region (in the UCSC I mean)?
The CpG island associated with your gene seems to be in the promoter and spanning the TSS, going into the transcribed region of the gene. It has a portion in the promoter and a portion in the gene. The promoter is located upstream of the gene, in the 5’ region adjacent to the transcription start site. In this case, the gene is read in direction to the left (indicated by the small arrows along the gene in the UCSC), so the promoter is the region to the right, in this case, adjacent to that 1st exon where the gene starts.