First pass at the requested dual-555 circuit.
R1-C1 is the 15 second power-on delay. R2-C2 is a differentiator to compensate for the 555's non-monostable behavior. It turns the U1 output trailing edge into a short trigger pulse. Its period is not critical, but must be less than the output pulse width. R3-C3 sets the 1 second output pulse time.
For both timers, the pulse width is 1.05 x R x C.
R4 provides a discharge path for C1 and C3 when power is removed. C4 and C5 are power supply decoupling caps for the two ICs.
The two ICs can have different power sources, but the two circuits must share a common GND.
Click on the schematic for a larger image.
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View attachment 147501