I quickly checked them, the network card seems to be detected by both kernels, so that's weird...Didi you have any time to check log files?
1e:00.0 Ethernet controller [0200]: Broadcom Inc. and subsidiaries NetXtreme BCM5723 Gigabit Ethernet PCIe [14e4:165b] (rev 10)
Subsystem: Hewlett-Packard Company NC107i Integrated PCI Express Gigabit Server Adapter [103c:705d]
Kernel driver in use: tg3
Kernel modules: tg3
1e:00.0 Ethernet controller [0200]: Broadcom Inc. and subsidiaries NetXtreme BCM5723 Gigabit Ethernet PCIe [14e4:165b] (rev 10)
Subsystem: Hewlett-Packard Company NC107i Integrated PCI Express Gigabit Server Adapter [103c:705d]
Kernel driver in use: tg3
Kernel modules: tg3
Do you need anything more?
I was a bit unclear regarding where the first two commands should be executed, so is the output of
ip addr
posted from the 5.4 or the 5.11 kernel?