How should I understand the pronunciation of 驾驶证 (jiàshǐ zhèng) as 酱汁儿 (jiàng zhī er)? I was reading this website ( https://pinyin.sogou.com/zimeiti/article/1255 ) to understand colloquial Beijing Mandarin and came across this pronunciation for 驾驶证

驾驶证 > "酱汁儿", i.e Jiàshǐ zhèng > jiàng zhī er

How should I understand these changes?

Is it something like 驾驶证儿, with shi disappearing in the middle (jia > jiang, shi> 0, zhengr > zhir), or is it 驾驶证 with shi becoming zhi and zheng becoming r (jia > jiang, shi> zhi, zheng > r)? I know ch, sh, and zh become r a lot but I also know whole syllables can disappear.
Your 1st interpretation is correct. In a three-character word the middle character tends to be ommitted.

驾驶证     jia shi zheng -&gt;
驾驶证儿   jia shi zh-r  -&gt;
驾儿证儿   jia r zh-r    -&gt;
驾(儿)证儿 jia-r zh-r    -&gt;
酱汁儿     jiang zh-r


This is the same for 派出所 becoming 派儿所 (盼儿锁 in the article).