Graph multiple geom_lines with varied timestamped data over multiple days

I have data collected over multiple days, with timestamps that contain information for when food was eaten. Example dataframes:

head(Day3)

==================================================================
Day3.time        Day3.Pellet_Count
1  18:05:30                 1
2  18:06:03                 2
3  18:06:34                 3
4  18:06:40                 4
5  18:06:52                 5
6  18:07:03                 6

head(Day4)

==================================================================
Day4.time Day4.Pellet_Count
1  18:00:21                 1
2  18:01:34                 2
3  18:02:22                 3
4  18:03:35                 4
5  18:03:54                 5
6  18:05:06                 6

Given the variability, the timestamps don't line up and therefore aren't matched. I've done a "full join" with merge from all of the data from two of the days, in the following way:

pellets <- merge(Day3, Day4, by = 'time', all=TRUE)

This results in the following:

head(pellets)

==================================================================
pellets.time         pellets.Pellet_Count.x   pellets.Pellet_Count.y
1     02:40:18                     39                     NA
2     18:00:21                     NA                      1
3     18:01:34                     NA                      2
4     18:02:22                     NA                      3
5     18:03:35                     NA                      4
6     18:03:54                     NA                      5

I would like to plot the Pellet_Count in one line graph from each of the days, but this is making it very difficult to group the data. My approach thus far has been:

pelletday <- ggplot() + geom_line(data=pellets, aes(x=time, y=Pellet_Count.x)) + geom_line(data=pellets, aes(x=time, y=Pellet_Count.y))

But, I get this error:

geom_path: Each group consists of only one observation. Do you need to adjust the group aesthetic?

I also would like to be able to merge all days (I oftentime have up to 9 days), and plot it on the same graph.

I believe my goal is to ultimately get the following dataframe output:

==================================================================
pellets.time                 Pellet_Count                Day
1     02:40:18                     39                     3
2     18:00:21                     1                      4
3     18:01:34                     2                      4
4     18:02:22                     3                      4
5     18:03:35                     4                      4
6     18:03:54                     5                      4

and to use this to graph: ggplot(pellets, aes(time, Pellet_Count, group=Day)

Any ideas?